## 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

log

_{2}(100) = log_{10}(100)/log_{10}(2) = 2 / 0.30103 = 6.64386## Example #2

log

_{3}(50) = log_{8}(50)/log_{8}(3) = 1.8812853 / 0.5283208 = 3.5608766## Proof

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

(1)

x=b^{log}b^{(x)}Raising c with the power of base c logarithm of b gives b:

(2)

b=c^{log}c^{(b)}When we take (1) and replace b with

c^{log}c^{(b)}(2), we get:(3)

x=b^{log}b^{(x)}= (c^{log}c^{(b)})^{log}b^{(x)}=c^{log}c^{(b)×log}b^{(x)}By applying log

() on both sides of (3):_{c}log

(_{c}x) = log(_{c}c^{log}c^{(b)×log}b^{(x)})By applying the logarithm power rule:

log

(_{c}x) = [log(_{c}b)×log(_{b}x)] ª log_{c}(c)Since log

_{c}(c)=1log

(_{c}x) = log(_{c}b)×log(_{b}x)Or

log

(_{b}x) = log(_{c}x)/log(_{c}b)

## See also