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:

logb(x) = logc(x) / logc(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 = blogb(x)

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

(2) b = clogc(b)

When we take (1) and replace b with clogc(b) (2), we get:

(3) x = blogb(x) = (clogc(b))logb(x) = clogc(b)×logb(x)

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

logc(x) = logc(clogc(b)×logb(x))

By applying the logarithm power rule:

logc(x) = [logc(b)×logb(x)] × logc(c)

Since logc(c)=1

logc(x) = logc(b)×logb(x)

Or

logb(x) = logc(x) / logc(b)

Logarithm of zero »


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