Logarithm/Identities

What follows is a list of logarithmic 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.

Change of base formula

log_ab = (log_cb)/(log_ca)

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

Note: these three identities lead to the use of logarithm tables 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

Special values

log_a(1) = 0

log_a(a) = 1

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

Derivative

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