Logarithmic identities

For people who check facts

What follows is a list of identities that are useful when dealing with logarithms. All of these are valid for all positive real numbers a, b and c except that the base of a logarithm may never be 1.

Table of contents

1 Special values
2 Multiplication, division and exponentiation
3 Logarithms and exponential functions are inverses
4 Change of base formula
5 Limits
6 Derivative
7 Integral

Special values

Multiplication, division and exponentiation

These three identities lead to the use of logarithm tables and slide rules; knowing the logarithm of two numbers allows you to multiply and divide them quickly, as well as compute powers and roots.

Logarithms and exponential functions are inverses

These are used to solve equations in which the unknowns occur in the exponent.

Change of base formula

This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log₁₀, but not for log₂. To find log₂(100), you have to calculate log₁₀(100) / log₁₀(2) (or ln(100)/ln(2), which is the same thing).

Limits

The last limit is often summarized as "logarithms grow more slowly than any power or root of x".

Derivative

Integral