Logarithm/Identities

< Logarithm

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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

logab = (logcb)/(logca)

Multiplication, division and exponentiation

logc(ab) = logca + logcb
logc(a/b) = logca - logcb
logc(ar) = r * logc(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

aloga(b) = b
loga (ar) = r     for all real numbers r

Special values

loga(1) = 0
loga(a) = 1

Limits

limx->0 loga(x) = -∞     if a > 1
limx->0 loga(x) = ∞     if a < 1


limx-> loga(x) = ∞     if a > 1
limx-> loga(x) = -∞     if a < 1


limx->0 loga(x) * xb = 0
limx-> loga(x) / xb = 0

Derivative

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