Logarithm Change of Base Rule

Logarithm change of base rule

In order to change base from b to c, we can use the logarithm change of base rule. The base b logarithm of x is equal to the base c logarithm of x divided by the base c logarithm of b:

log_b(x) = log_c(x) / log_c(b)

Example #1

log₂(100) = log₁₀(100) / log₁₀(2) = 2 / 0.30103 = 6.64386

Example #2

log₃(50) = log₈(50) / log₈(3) = 1.8812853 / 0.5283208 = 3.5608766

Proof

Raising b with the power of base b logarithm of x gives x:

(1) x = b^logb^(x)

Raising c with the power of base c logarithm of b gives b:

(2) b = c^logc^(b)

When we take (1) and replace b with c^logc^(b) (2), we get:

(3) x = b^logb^(x) = (c^logc^(b))^logb^(x) = c^logc^(b)×logb^(x)

By applying log_c() on both sides of (3):

log_c(x) = log_c(c^logc^(b)×logb^(x))

By applying the logarithm power rule:

log_c(x) = [log_c(b)×log_b(x)] × log_c(c)

Since log_c(c)=1

log_c(x) = log_c(b)×log_b(x)

log_b(x) = log_c(x) / log_c(b)

Logarithm of zero ►

Logarithm Change of Base Rule