How to use the LOG10 function
What is the LOG10 function?
The LOG10 function calculates the logarithm of a number using the base 10.
Table of Contents
1. Introduction
What is a logarithm?
In logarithms, the base is the number that is raised to a power to produce the desired output. It is the foundation of the logarithmic function.
What is the log10?
The logarithm base 10, or log10, also known as the common logarithm, is the special case where the base is 10:
log10(x) = y where x and y are numbers such that 10^y = x
In other words, log10(x) is the power that 10 must be raised to in order to equal x.
What is the definition of a logarithm with an base of 10?
For the base 10:
log10(x) = lim (n->infinity) (x^(1/n) - 1) * n / log10(e)
Where:
-
- 10 is the base
- x is the input number
- e is the mathematical constant (approximately 2.71828...)
This limit definition comes from Euler's Logarithm Definition, relating logs of any base b to the natural log ln(x).
What is a base?
In logarithms, the base is the number that is raised to a power to produce the desired output. It is the foundation of the logarithmic function.
For example, the logarithm log10(100):
- 10 is the base
- 100 is the input number
- 2 is the exponent that makes 10 return 100 because 102 = 100
The most common bases are
- 10
- e (natural log), and
- 2 (for computers).
But any positive number besides 1 can be a base.
What are the four key logarithm rules?
The following rules are essential to learn if you want to solve various equations involving natural logarithms effectively.
- Product rule
log10(x*y) = log10(x) + log10(y)
The log10 of the multiplication of x and y is the sum of the log10 x and log10 y.
Example, log10(5*10) = log10(5) + log10(10) - Quotient rule
log10(x/y) = log10(x) - log10(y)
The log10 of the division of x and y is the difference of the log10 x and log10 y.
Example, log10(5/10) = log10(5) - log10(10) - Reciprocal rule
log10(1/x) = − log10(x)
The natural log of the reciprocal of x is similar to the quotient rule. log10(1/x) = log10(1) - log10(x) = 0 - log10 x = - log10 x
Example, log10(1/5) = - log10(5) - Power rule
log10(xy) = y * log10(x)
The log10 of x raised to the power of y is y multiplied by the log10 x.
Example, log10(510) = 10 * log10(5)
What are the other logarithmic functions in Excel?
| Excel function | Description |
|---|---|
| LOG | Returns the logarithm with a given base. |
| LN | Returns the natural logarithm (base e) of a number |
| LOG10 | Returns the base-10 logarithm of a number |
| LOG2 | Returns the base-2 logarithm of a number |
The corresponding inverse functions for the logarithm functions in Excel:
| Logarithm | Power to |
|---|---|
| LOG | Arbitrary base, basex or POWER(number, power) |
| LN | EXP(number) |
| LOG10 | 10x or POWER(10,x) |
| LOG2 | 2x or POWER(2,x) |
2. Syntax
LOG10(number)
| number | Required. A value larger than 0 (zero) that you want to calculate the logarithm with base 10. |
3. Example 1
This example demonstrates how to use the log10 function, the image above shows a chart with different logarithmic bases. Below the chart is a table containing numbers (B25:B29) and next to the numbers are the log10 calculations in cell range D25:D29.
log10(x) = y where x and y are numbers so that 10y = x
The first number in cell B25 is 1. The output from the log10 function is 0 (zero).
Formula in cell C25:
If we calculate 100 we get 1. This matches the number in cell B25. To calculate 10 to the power of 0 (zero) in Excel we can use the caret character ^, here is how: =10^0 This will calculate 10 to the power of 0 (zero) in an Excel formula. We can also use the POWER function: =POWER(number, power) For example, POWER(10,0) which returns 1.
We can also use the chart to verify the values. Find 1 on the x-axis, follow an imaginary vertical line up until you intersect the log10 curve. In this case, it is exactly 0 (zero) on the y-axis located on the left side of the chart.
The second number is in cell B26 and it is 10.
Formula in cell C26:
The output from the log10 function is 1. If we calculate 101 we get 10. This matches the number in cell B26.
We can also use the chart to verify the values. Find 10 on the x-axis, follow an imaginary vertical line up until you intersect the log10 curve. Then follow an imaginary horizontal line to the left until you reach the y-axis. In this case, it is exactly 1 on the y-axis located on the left side of the chart.
The third number is in cell B27 and it is 100.
Formula in cell C27:
The output from the log10 function is 2. If we calculate 102 we get 100. This matches the number in cell B27.
The fourth number is in cell B28 and it is 45.
Formula in cell C28:
The output from the log10 function is approx. 1.653. If we calculate 101.65321251377534 we get 45. This matches the number in cell B28.
The fourth number is in cell B29 and it is 33.
Formula in cell C28:
The output from the log10 function is approx. 1.5185. If we calculate 101.51851393987789 we get 33. This matches the number in cell B28.
4. Example 2
An initial seismic event registered a magnitude of 4.0 on the Richter scale. A subsequent earthquake was 900 times greater than the initial event. Given that the Richter scale is logarithmic (base 10) and that each whole number increase represents a 10-fold increase in ground motion amplitude , determine the Richter magnitude of the second, more powerful earthquake?
Formula in cell D23:
LOG10(900) evaluates to approx 2.9542, add 4 to this number and we get the magnitude for the subsequent earthquake which is approx. 6.9542
The image shows a graph representing the Richter scale for measuring earthquake intensity. The x-axis represents the Richter scale magnitude, ranging from 1 to 10. The y-axis shows a logarithmic scale for seismic amplitude, ranging from 1 to 10^10 (10 billion). The scale is logarithmic, meaning each major grid line represents a 10-fold increase. The blue dots connected by lines show the exponential relationship between Richter magnitude and seismic amplitude.
A green horizontal line is drawn at around the 10,000 level on the y-axis, corresponding to a magnitude of approximately 4 on the Richter scale (labeled as the "First earth quake" with a value of 4).
A red vertical line extends from around magnitude 6.9 on the x-axis up to the curve, then horizontally to the y-axis, intersecting at about 9,000,000 on the logarithmic scale.
5. LOG10 not working
The LOG10 function returns #NUM! error if number is equal to 0 (zero) or smaller.
7. How to raise a number to the power of
The POWER function lets you raise a number to the power of a specific value. For example, =POWER(10,2) is the same as 102 equals 100.
The caret character lets you also raise a number to the power of a specific value, it is smaller than the POWER function which then makes your formulas smaller. For example, =10^2 is the same as =POWER(10,2) or 102 which equals 100.
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Functions in 'Math and trigonometry' category
The LOG10 function function is one of 62 functions in the 'Math and trigonometry' category.
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