Logarithmic identities

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

log_a(1) = 0

log_a(a) = 1

Multiplication, division and exponentiation

log_c(ab) = log_ca + log_cb

log_c(a/b) = log_ca - log_cb

log_c(a^r) = r log_c(a) for all real numbers r

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

a^log_a(b) = b

log_a (a^r) = r for all real numbers r

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

Change of base formula

log_ab = (log_cb)/(log_ca)

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

lim_x->0 log_a(x) = -∞ if a > 1

lim_x->0 log_a(x) = ∞ if a < 1

lim_x->∞ log_a(x) = ∞ if a > 1

lim_x->∞ log_a(x) = -∞ if a < 1

lim_x->0 log_a(x) * x^b = 0

lim_x->∞ log_a(x) / x^b = 0

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

Derivative

d/dx log_a(x) = 1 / (x ln(a))

Integral

∫ log_a(x) = x log_a(x) - log_a(x)