How to Put Base Log on Calculator

Calculating logarithms with different bases is a common requirement in mathematics, science, and engineering. This guide explains how to perform base log calculations on your calculator, including the change of base formula and practical examples.

What is Base Log?

A logarithm is the inverse operation to exponentiation. While a base-10 logarithm (log₁₀) answers "to what power must 10 be raised to obtain the number," a base log (logₐ) answers "to what power must a (where a ≠ 1) be raised to obtain the number."

Most scientific calculators have a log button that calculates base-10 logarithms. However, if you need to calculate logarithms with a different base, you'll need to use the change of base formula.

How to Calculate Base Log

To calculate a logarithm with a different base than your calculator's default, follow these steps:

Identify the number (N) you want to find the logarithm of
Identify the desired base (a)
Use the change of base formula: logₐ(N) = log(N)/log(a)
Calculate log(N) using your calculator's log button
Calculate log(a) using your calculator's log button
Divide the two results to get logₐ(N)

Note: This method works for any positive real number N and base a, where a ≠ 1 and N ≠ 0.

Using Your Calculator

Most scientific calculators have a log button that calculates base-10 logarithms. To calculate logarithms with a different base:

Enter the number you want to find the logarithm of
Press the log button to calculate log₁₀(N)
Enter the base you want to use
Press the log button to calculate log₁₀(a)
Divide the first result by the second result to get logₐ(N)

For example, to calculate log₂(8):

Enter 8 and press log: result is approximately 0.9031
Enter 2 and press log: result is approximately 0.3010
Divide 0.9031 by 0.3010: result is approximately 3

Change of Base Formula

The change of base formula allows you to calculate logarithms with any base using your calculator's default logarithm function:

logₐ(N) = log(N)/log(a)

Where:

logₐ(N) is the logarithm of N with base a
log(N) is the logarithm of N with your calculator's default base (usually base-10)
log(a) is the logarithm of a with your calculator's default base

This formula works because logarithms with different bases are proportional to each other.

Worked Examples

Example 1: log₂(16)

Using the change of base formula:

log₂(16) = log(16)/log(2)

log(16) ≈ 1.2041

log(2) ≈ 0.3010

log₂(16) ≈ 1.2041 / 0.3010 ≈ 4

This makes sense because 2⁴ = 16.

Example 2: log₅(125)

Using the change of base formula:

log₅(125) = log(125)/log(5)

log(125) ≈ 2.0969

log(5) ≈ 0.6990

log₅(125) ≈ 2.0969 / 0.6990 ≈ 3

This makes sense because 5³ = 125.

FAQ

Can I use the change of base formula with any base?: Yes, the change of base formula works for any positive real number base (a) where a ≠ 1.
What happens if I try to calculate log₁(N)?: Logarithms with base 1 are undefined because 1 raised to any power is always 1, and there's no solution to the equation 1^x = N for N ≠ 1.
Can I use the change of base formula with natural logarithms?: Yes, you can use the change of base formula with natural logarithms (ln) by replacing log with ln in the formula: logₐ(N) = ln(N)/ln(a).
What if my calculator doesn't have a log button?: If your calculator doesn't have a log button, you can still calculate logarithms using the change of base formula by first calculating the natural logarithm (ln) of the numbers.
How accurate are the results from the change of base formula?: The accuracy depends on the precision of your calculator. Most scientific calculators provide at least 10 decimal places of accuracy.