Choose what you want to find from the menu, enter the base a and the exponent k, and press "Calculate". The answer is shown two ways, by hand with a given approximation (such as log₁₀ 2 ≈ 0.3010) and exactly from the full value.
In the number of digits and leading digit modes, a is a whole number from 2 to 1000000. In the decimal place mode, a is a decimal greater than 0 and less than 1 (up to 6 decimal places). The exponent k is a whole number from 1 to 10000.
Result and graph
Enter the base and the exponent in the fields on the left and press "Calculate". The result and a number line of powers of 10 will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the base \(a\) and the exponent \(k\), and find on the spot how many digits a huge power \(a^k\) such as \(2^{100}\) has
- You get two answers: one by hand with a given approximation such as \(\log_{10} 2 \approx 0.3010\) (the method in Algebra 2 and precalculus textbooks), and the exact number of digits from calculating the whole value. The steps follow the same order as a written solution
- Another mode finds the leading digit (the leftmost digit) from the fractional part of the common logarithm
- For a power of a decimal less than 1, such as \(0.5^{100}\), it finds the decimal place where the first nonzero digit appears
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
This page handles powers of whole numbers (number of digits and leading digit) and powers of terminating decimals less than 1 (decimal place of the first nonzero digit).
What is this calculation used for?
Estimating the size of an encryption key in digits (IT and security)
RSA encryption, which protects communication on the internet, uses keys described as "2048-bit". A 2048-bit number is about \(2^{2048}\) in size, and from \(2048 \times \log_{10} 2 \approx 616.5\) it has 617 decimal digits.
The conversion "number of bits × 0.3010 ≈ number of decimal digits" is exactly the calculation on this page. This estimate shows how unrealistic it is to break such a key by trying every possibility.
Bacterial growth and preventing food poisoning (food safety)
Some bacteria divide about once every 20 minutes under good conditions. If that pace lasted 24 hours, there would be 72 divisions, and 1 bacterium would become \(2^{72}\). From \(72 \times 0.3010 \approx 21.7\), that is a 22-digit number (about \(4.7 \times 10^{21}\)).
In reality, limits on food and space stop the growth at some point. Still, one multiplication with the common logarithm shows that bacteria left alone multiply by many powers of ten. That is why the rule is to refrigerate food quickly.
Could folding paper reach the Moon? (a feel for exponential growth)
Each time you fold a sheet of paper 0.004 inch (about 0.1 mm) thick, its thickness doubles. If you could fold it 42 times, it would be \(0.004 \times 2^{42}\) inches \(\approx\) 278,000 miles thick, farther than the Moon (about 239,000 miles away).
In reality, paper can only be folded about 7 or 8 times, so this is a thought experiment. Even so, how fast the digits grow when you keep doubling can be predicted exactly by the simple formula \(k \times \log_{10} 2\), and you can check your intuition with numbers.
Detecting fraud in accounting data (Benford's law)
In naturally collected money data, the leading digit is 1 about 30.1% of the time (equal to \(\log_{10} 2\)), 2 about 17.6% of the time, and so on. This lopsided pattern is known as Benford's law.
Made-up numbers tend to break this pattern, so auditors and fraud investigators use it as a check. The idea on this page, "the leading digit is decided by the fractional part of the common logarithm", is the basis of the law.
The language of powers of ten in science (science and astronomy)
Science writes numbers that are too large or too small as powers of 10 (scientific notation), such as the distance to the Sun, about 1.496×10⁸ km, or the size of a hydrogen atom, about 1×10⁻¹⁰ m.
The exponent of 10 is exactly the integer part of the common logarithm, a direct sense of the number of digits. Astronomy and chemistry (Avogadro's number is about 6×10²³) are great places to practice thinking in powers of ten.
Formulas and figures
Common logarithm of a power (move the exponent to the front)
Standard notation
(the usual math form)
\(\log_{10} a^{k}\)
\(=\)
\(k\)
\(\times\)
\(\log_{10} a\)
In words
(symbols replaced with words)
③
\(\log_{10} a^k\): common log of the power
\(=\)
①
\(k\): exponent
\(\times\)
②
\(\log_{10} a\): common log of the base
The formula in words
①
Multiply the
\(k\): exponent
②
by the
\(\log_{10} a\): common log of the base \(a\)
③
and you get the
\(\log_{10} a^k\): common log of the power \(a^k\)
Quick example
Using \(\log_{10} 2 \approx 0.3010\), the common log of \(2^{100}\) is
common log of \(2^{100}\)
\(=\)
exponent 100
\(\times\)
common log of 2 (about 0.3010)
\(\log_{10} 2^{100} = 100 \times \log_{10} 2 \approx 100 \times 0.3010 = 30.10\)
Key idea
Finding \(2^{100}\) by actual multiplication takes 99 multiplications. With the common logarithm (the logarithm with base 10), a single multiplication, "exponent × \(\log_{10} 2\)", tells you the rough size of the number (about 10 to what power). Logarithms do the repeated multiplication for you. That is what they do best.
\(\log_{10} 2 = 0.30102\ldots\) goes on forever, so by hand you use an approximation to 4 decimal places, such as 0.3010. Textbook problems usually give it to you, as in "use \(\log_{10} 2 \approx 0.3010\)".
Number of digits (\(10^{n-1} \le N < 10^n\) ⇔ \(n\) digits)
Figure (number line of powers of 10)
Standard notation
(the usual math form)
\(10^{n-1}\)
\(\le\)
\(N\)
\(<\)
\(10^{n}\)
In words
(symbols replaced with words)
②
\(10^{n-1}\): smallest \(n\)-digit number
\(\le\)
①
\(N\): number checked
\(<\)
③
\(10^{n}\): smallest \((n+1)\)-digit number
The formula in words
①
When the whole number
\(N\): number checked
is at least the
②
\(10^{n-1}\): smallest \(n\)-digit number (1 followed by \(n-1\) zeros)
and less than the
③
\(10^{n}\): smallest \((n+1)\)-digit number
, \(N\) has exactly \(n\) digits
Quick example
From \(\log_{10} 2^{100} \approx 30.10\), we get \(2^{100} \approx 10^{30.10}\), so
smallest 31-digit number \(10^{30}\)
\(\le\)
number checked \(2^{100}\)
\(<\)
smallest 32-digit number \(10^{31}\)
\(30 \le 30.10 < 31\)
\(10^{30} \le 2^{100} < 10^{31}\)
Key idea
1-digit numbers are 1 to 9 (\(10^0 \le N < 10^1\)), 2-digit numbers are 10 to 99 (\(10^1 \le N < 10^2\)), and so on. Each extra digit raises the exponent of 10 by 1. So if \(N\) is between \(10^{n-1}\) and \(10^n\), it has \(n\) digits.
In the language of common logarithms, "the number of digits \(n\) is the integer part of \(\log_{10} N\) plus 1". The integer part of \(\log_{10} 2^{100} \approx 30.10\) is 30, so \(2^{100}\) has 31 digits.
Leading digit (from the fractional part of the common log)
Standard notation
(the usual math form)
\(\log_{10} d\)
\(\le\)
\(f\)
\(<\)
\(\log_{10} (d+1)\)
In words
(symbols replaced with words)
②
\(\log_{10} d\): common log of the leading digit
\(\le\)
①
\(f\): fractional part of \(\log_{10} a^k\)
\(<\)
③
\(\log_{10} (d+1)\): common log of the next digit
The formula in words
①
When the
\(f\): fractional part of the common log
is at least the
②
\(\log_{10} d\): common log of the leading digit \(d\)
and less than the
③
\(\log_{10} (d+1)\): common log of the next digit \(d+1\)
, the leading digit is \(d\)
Quick example
The fractional part of \(\log_{10} 2^{100} \approx 30.10\) is \(0.10\). It lies between \(\log_{10} 1 = 0\) and \(\log_{10} 2 \approx 0.3010\), so
common log of 1 (0)
\(\le\)
fractional part 0.10
\(<\)
common log of 2 (about 0.3010)
\(0 \le 0.10 < 0.3010\)
\(1 \le 10^{0.10} < 2\)
Key idea
Split it as \(2^{100} \approx 10^{30.10} = 10^{30} \times 10^{0.10}\). The integer part, \(10^{30}\), only decides how many zeros follow (where the digits sit). It does not change the digits themselves. The first digit is decided by the fractional part, \(10^{0.10}\).
Since \(1 \le 10^{0.10} < 2\), \(2^{100}\) has the form \(1.\ldots \times 10^{30}\), so its leading digit is 1. To decide, compare the fractional part with the "scale of common logs of 1 to 9": \(\log_{10} 1 = 0\), \(\log_{10} 2 \approx 0.3010\), \(\log_{10} 3 \approx 0.4771\), and so on.
Decimals less than 1 (\(10^{-m} \le N < 10^{-m+1}\) ⇔ decimal place \(m\))
Standard notation
(the usual math form)
\(10^{-m}\)
\(\le\)
\(N\)
\(<\)
\(10^{-m+1}\)
In words
(symbols replaced with words)
②
\(10^{-m}\): first 1 at decimal place \(m\)
\(\le\)
①
\(N\): number checked
\(<\)
③
\(10^{-m+1}\): 10 times that number
The formula in words
①
When a number less than 1, the
\(N\): number checked
, is at least the
②
\(10^{-m}\): number whose first 1 is at decimal place \(m\) (in the form 0.00…01)
and less than
③
\(10^{-m+1}\): 10 times that number
, the first nonzero digit of \(N\) appears at decimal place \(m\)
Quick example
Since \(\log_{10} 0.5^{100} \approx 100 \times (-0.3010) = -30.10\),
first 1 at decimal place 31 \(10^{-31}\)
\(\le\)
number checked \(0.5^{100}\)
\(<\)
10 times that \(10^{-30}\)
\(-31 \le -30.10 < -30\)
\(10^{-31} \le 0.5^{100} < 10^{-30}\)
Key idea
For numbers less than 1, the common logarithm is negative (\(\log_{10} 0.1 = -1\), \(\log_{10} 0.01 = -2\), and so on). The idea is the same as for the number of digits: just put \(N\) between two powers of 10.
Be careful with "the integer part of a negative number". Here the integer part is the floor, the nearest whole number to the left on the number line, so the integer part of \(-30.10\) is \(-31\), not \(-30\). Getting this wrong makes the answer off by 1. In this example, \(-31 \le -30.10 < -30\), so the first nonzero digit of \(0.5^{100}\) appears at the 31st decimal place.
A whole number \(N\) has \(n\) digits when \(10^{n-1} \le N < 10^n\). So the number of digits of \(a^k\) is just the integer part of the common logarithm \(\log_{10} a^k = k \log_{10} a\) plus 1. The fractional part gives the leading digit, and for a number less than 1 the same idea tells you the decimal place where the first nonzero digit appears.
Symbols and terms
Symbols
| \(\log_{10} N\) | log base 10 of N | The common logarithm of \(N\) (the logarithm with base 10). It tells "10 to what power gives \(N\)". log is short for logarithm. |
| \(N\) | N | The number you are checking, for example for its number of digits. It is the first letter of "number". On this page it is the power \(a^k\). |
| \(a\) | a | The base (the number being multiplied). By convention, letters from the start of the alphabet are used for fixed numbers. |
| \(k\) | k | The exponent (how many times to multiply). In \(2^{100}\), it is 100. |
| \(n\) | n (lowercase) | The number of digits. It is the first letter of "number" and is often used for counts. |
| \(10^n\) | 10 to the nth power | 10 multiplied \(n\) times: 1 followed by \(n\) zeros (\(10^3 = 1000\)). It is the smallest \((n+1)\)-digit number. |
| \(f\) | f | The fractional part of the common logarithm, from the first letter of "fractional". The leading digit depends only on this \(f\). |
| \(d\) | d | The leading digit (the leftmost digit), from the first letter of "digit". |
| \(m\) | m | The decimal place where the first nonzero digit appears. It shows "how small" a number less than 1 is. |
| \(\lfloor x \rfloor\) | floor of x | The floor function: the greatest integer not greater than \(x\) (the integer part). Be careful with negative numbers: \(\lfloor -30.10 \rfloor = -31\). |
Terms
| common logarithm (common log) | The logarithm with base 10, \(\log_{10} x\). It tells "10 to what power gives \(x\)" and is tied directly to the number of digits in base 10. It is taught in Algebra 2. |
| logarithm | The number that tells what power of the base gives the argument. It is exactly the reverse of a power. |
| argument | The number inside a logarithm, the \(N\) in \(\log_{10} N\). Only numbers greater than 0 can be used. |
| base | The number \(a\) that a power \(a^k\) or a logarithm \(\log_{a} x\) is built on. On this page it is the "Base a" field. The common logarithm has base 10. |
| number of digits | How many digits appear when you write a whole number. 1024 has 4 digits. In math terms, "a whole number that is at least \(10^{n-1}\) and less than \(10^n\) has \(n\) digits". |
| leading digit | The leftmost digit of a number. The leading digit of 1024 is 1. It can be found from the fractional part of the common logarithm. |
| power | Multiplying the same number by itself several times. As in \(2^{3} = 2 \times 2 \times 2\), the small exponent at the upper right tells how many times. |
| exponent | The small number at the upper right of a power that tells what power to raise to. In \(2^{100}\), the exponent is 100. |
| integer part (characteristic) | When you split a number into "the greatest integer not greater than it" (the floor) and "the rest", this is the integer. For 30.10 it is 30. Note that for \(-30.10\) it is \(-31\), because the fractional part is always taken to be at least 0 and less than 1. For a common logarithm, it is also called the characteristic. |
| fractional part (mantissa) | When you split a number into "the greatest integer not greater than it" and "the rest", this is the rest. For 30.10 it is 0.10. The fractional part is always at least 0 and less than 1, so for \(-30.10\) it is \(0.90\). For a common logarithm, it is also called the mantissa. |
| approximation | A rounded value close to the true value that is easier to work with, such as using 0.3010 for \(\log_{10} 2 = 0.30102\ldots\). Rounding leaves an error, and multiplication makes the error larger. |
| significant figures | The digits of a measurement or approximation that can be trusted. 0.3010 is accurate only to 4 decimal places, so the fine detail of 10000 times it cannot be trusted. This explains why the answer by hand can differ from the exact answer on this page. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to these topics is the fastest way forward.
| Powers and exponents (Grades 5–6) |
|
| Extending exponents (Grade 8 and Algebra 2) |
|
| Definition of a logarithm (Algebra 2) |
|
| Log rules (Algebra 2) |
|
| Reading and writing inequalities (Grades 6–7 and Algebra 1) |
|
| Place value and digits (Grade 4 and up) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the number of digits of a^k
| Base a | 2 |
| Exponent k | 100 |
| Common log k×log10(a) | =B2*LOG10(B1) |
| Number of digits (integer part + 1) | =INT(B3)+1 |
Table to find the leading digit
| Base a | 2 |
| Exponent k | 100 |
| Common log k×log10(a) | =B2*LOG10(B1) |
| Fractional part of the common log | =B3-INT(B3) |
| Leading digit | =INT(10^B4) |
Table to find the first nonzero decimal place of a^k (a < 1)
| Base a (a decimal less than 1) | 0.5 |
| Exponent k | 100 |
| Common log k×log10(a) | =B2*LOG10(B1) |
| Decimal place of the first nonzero digit | =-INT(B3) |
After pasting, the upper rows (base and exponent) are your inputs, and the lower rows show the results calculated automatically.
LOG10 is the common logarithm, and INT gives the greatest integer not greater than the number (the integer part, or floor). INT rounds down even for negative numbers (INT(-30.1) is -31), so "=-INT(B3)" in the third table gives the decimal place directly.
With the example numbers, the first table shows 31 (digits), the second shows 1 (the leading digit), and the third shows 31 (the 31st decimal place).
Excel calculates with about 15 digits of precision, so when the exponent is very large the answer can be off by 1 near a power of 10.
LOG10 is the common logarithm, and INT gives the greatest integer not greater than the number (the integer part, or floor). INT rounds down even for negative numbers (INT(-30.1) is -31), so "=-INT(B3)" in the third table gives the decimal place directly.
With the example numbers, the first table shows 31 (digits), the second shows 1 (the leading digit), and the third shows 31 (the 31st decimal place).
Excel calculates with about 15 digits of precision, so when the exponent is very large the answer can be off by 1 near a power of 10.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the number of digits of a^k
| Base a | 2 |
| Exponent k | 100 |
| Common log k×log10(a) | =B2*LOG10(B1) |
| Number of digits (integer part + 1) | =INT(B3)+1 |
Table to find the leading digit
| Base a | 2 |
| Exponent k | 100 |
| Common log k×log10(a) | =B2*LOG10(B1) |
| Fractional part of the common log | =B3-INT(B3) |
| Leading digit | =INT(10^B4) |
Table to find the first nonzero decimal place of a^k (a < 1)
| Base a (a decimal less than 1) | 0.5 |
| Exponent k | 100 |
| Common log k×log10(a) | =B2*LOG10(B1) |
| Decimal place of the first nonzero digit | =-INT(B3) |
Google Sheets has the same LOG10 and INT functions as Excel, so exactly the same formulas work.
Copy the whole table, paste it into cell A1, and replace the base and exponent with your own numbers.
Copy the whole table, paste it into cell A1, and replace the base and exponent with your own numbers.
How to calculate it in Python
import math
a = 2 # base (a whole number 2 or more)
k = 100 # exponent (the power)
# Using the common logarithm (the same idea as the method by hand)
log_value = k * math.log10(a)
digit_count = math.floor(log_value) + 1 # integer part + 1 = number of digits
fractional = log_value - math.floor(log_value)
leading_digit = math.floor(10 ** fractional) # leading digit from the fractional part
print(f"Common log: {log_value}")
print(f"Number of digits: {digit_count}, leading digit: {leading_digit}")
# Exact calculation (Python integers have no digit limit, so we can check by calculating the whole value)
actual = a ** k
print(f"Actual number of digits: {len(str(actual))}, actual leading digit: {str(actual)[0]}")
# For a decimal less than 1 (which decimal place for 0.5 to the 100th power), the common log is negative
position = -math.floor(100 * math.log10(0.5))
print(f"The first nonzero digit of 0.5 to the 100th power appears at decimal place {position}")
Runs with only the math module from the standard library. The key point is to use math.floor(), not int(), for the integer part (int() rounds toward 0, so with a negative logarithm the answer would be off by 1). Change a and k at the top to your own numbers and run it.
How to write it in LaTeX and other math languages (copy and paste)
Common logarithm of a power (move the exponent to the front)
log₁₀(aᵏ) = k log₁₀ a
\log_{10} a^{k} = k \log_{10} a
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>log</mi><mn>10</mn></msub>
<msup><mi>a</mi><mi>k</mi></msup>
<mo>=</mo>
<mi>k</mi>
<msub><mi>log</mi><mn>10</mn></msub>
<mi>a</mi>
</mrow>
</math>
log_10 (a^k) = k log_10 a
Log10[a^k] == k*Log10[a]
log10(a^k) = k*log10(a);
log10(a^k) == k*log10(a)
log_10(a^k) = k log_10(a)
Number of digits (\(10^{n-1} \le N < 10^n\) ⇔ \(n\) digits)
10ⁿ⁻¹ ≤ N < 10ⁿ
10^{n-1} \le N < 10^{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mn>10</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup>
<mo>≤</mo>
<mi>N</mi>
<mo><</mo>
<msup><mn>10</mn><mi>n</mi></msup>
</mrow>
</math>
10^(n-1) <= N < 10^n
IntegerLength[a^k]
length(a^k);
floor(log10(a^k)) + 1
10^(n-1) ≤ N < 10^n
Leading digit (from the fractional part of the common log)
log₁₀ d ≤ f < log₁₀(d+1)
\log_{10} d \le f < \log_{10} (d+1)
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub><mi>log</mi><mn>10</mn></msub>
<mi>d</mi>
<mo>≤</mo>
<mi>f</mi>
<mo><</mo>
<msub><mi>log</mi><mn>10</mn></msub>
<mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo>
</mrow>
</math>
log_10 d <= f < log_10 (d+1)
First[IntegerDigits[a^k]]
trunc(10^frac(k*log10(a)));
floor(10^(mod(k*log10(a), 1)))
log_10(d) ≤ f < log_10(d+1)
Decimals less than 1 (\(10^{-m} \le N < 10^{-m+1}\) ⇔ decimal place \(m\))
10⁻ᵐ ≤ N < 10⁻ᵐ⁺¹
10^{-m} \le N < 10^{-m+1}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mn>10</mn><mrow><mo>−</mo><mi>m</mi></mrow></msup>
<mo>≤</mo>
<mi>N</mi>
<mo><</mo>
<msup><mn>10</mn><mrow><mo>−</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup>
</mrow>
</math>
10^(-m) <= N < 10^(-m+1)
-Floor[Log10[a^k]]
-floor(log10(a^k));
-floor(log10(a^k))
10^(-m) ≤ N < 10^(-m+1)
How to have ChatGPT do the calculation
You are a calculation assistant for common logarithms and the number of digits. Do the following calculations by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). 1. How many digits does 2 to the 100th power have? Find it both by adding 1 to the integer part of k × log10(a) and by counting directly with len(str(2**100)), and check that they agree. 2. Find the leading digit (the leftmost digit) of 2 to the 100th power from the fractional part of the common logarithm, and compare it with the actual value. 3. At which decimal place does the first nonzero digit of 0.5 to the 100th power appear? Find it with -floor(100 × log10(0.5)). Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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