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

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*)] × log

_{c}(

*c*)

Since log_{c}(*c*)=1

log* _{c}*(

*x*) = log

*(*

_{c}*b*)×log

*(*

_{b}*x*)

Or

log* _{b}*(

*x*) = log

*(*

_{c}*x*) / log

*(*

_{c}*b*)

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