How to Put A Log Base in Calculator

Logarithms with different bases are essential in mathematics, science, and engineering. This guide explains how to calculate logarithms with custom bases using a calculator, including step-by-step instructions and practical examples.

What is a Log Base?

A logarithm with a specified base is a mathematical function that answers the question: "To what power must the base be raised to obtain the given number?" The general form is:

log_b(x) = y means b^y = x

Where:

b is the base (must be positive and not equal to 1)
x is the number whose logarithm is being calculated (must be positive)
y is the result (the logarithm)

For example, log₂(8) = 3 because 2³ = 8.

How to Calculate Logarithms with Different Bases

To calculate a logarithm with a specific base, you can use the change of base formula:

log_b(x) = log_k(x) / log_k(b)

Where k is any positive number (commonly 10 or e for calculators). This formula allows you to use your calculator's built-in logarithm functions (usually base 10 or natural logarithm) to compute logarithms with any base.

Step-by-Step Calculation

Identify the base (b) and the number (x) for which you want to calculate the logarithm.
Use your calculator to find log_k(x) and log_k(b).
Divide the result from step 2 by the result from step 3.
The result is log_b(x).

Note: Most scientific calculators have a "log" button for base 10 and a "ln" button for natural logarithm (base e). Use these functions for k in the change of base formula.

Using a Calculator for Log Base

Here's how to use a standard calculator to compute logarithms with different bases:

Example Calculation

Calculate log₃(27):

Press the "log" button (base 10) and enter 27. The calculator shows approximately 1.4314.
Press the "log" button again and enter 3. The calculator shows approximately 0.4771.
Divide the first result by the second result: 1.4314 / 0.4771 ≈ 3.
The result is log₃(27) = 3.

This confirms that 3³ = 27, which is correct.

Common Logarithm Bases

While you can use any positive base (except 1) in logarithms, some bases are particularly common:

Base 10 (Common Logarithm): Used in many scientific and engineering applications, especially when dealing with powers of 10.
Base e (Natural Logarithm): Used in calculus and exponential growth/decay problems, where e ≈ 2.71828.
Base 2 (Binary Logarithm): Used in computer science and information theory, particularly when dealing with binary numbers.

For most practical purposes, you can use the change of base formula to convert between these common bases.

Worked Examples

Example 1: Base 5 Logarithm

Calculate log₅(125):

log₁₀(125) ≈ 2.0969
log₁₀(5) ≈ 0.6990
2.0969 / 0.6990 ≈ 3

Result: log₅(125) = 3 (since 5³ = 125)

Example 2: Base e Logarithm

Calculate log_e(7.389):

ln(7.389) ≈ 2 (since e² ≈ 7.389)
ln(e) = 1
2 / 1 = 2

Result: log_e(7.389) = 2

FAQ

What is the difference between log and ln?: The "log" function typically refers to base 10 logarithms, while "ln" refers to natural logarithms (base e). Both can be calculated using the change of base formula.
Can I use any base for logarithms?: Yes, you can use any positive base except 1. The base must be greater than 0 and not equal to 1.
How do I calculate logarithms with a base that isn't 10 or e?: Use the change of base formula: log_b(x) = log_k(x) / log_k(b), where k is 10 or e (whichever your calculator supports).
What happens if I try to calculate log_b(x) where x is negative?: Logarithms are only defined for positive real numbers. Attempting to calculate log_b(x) where x ≤ 0 will result in an error or undefined value.
How can I verify my logarithm calculations?: You can verify by raising the base to the power of the logarithm result. For example, if log₃(27) = 3, then 3³ should equal 27.