# Change of Base Formula

How to use the change of base formula to solve logarithmic problems: formula, proof, examples, and their solutions.

## Formula

log_{b} *c* = (log_{a} *c*) / (log_{a} *b*)*a* is the base.

So 0 < *a* < 1, *a* > 1.

## Proof

Set *b*^{(brown)} = *c*.

Log both sides. (base: *b*)

(brown) = log_{b} *c*

Logarithmic form

See *b*^{(brown)} = *c* again.

Log both sides. (base: *a*)

log_{a} *b*^{(brown)} = log_{a} *c*

(brown) = (log_{a} *c*) / (log_{a} *b*)

And (brown) = log_{b} *c*.

So log_{b} *c* = (log_{a} *c*) / (log_{a} *b*).

## Example 1

The given conditions are common logs. (base 10)

So change the base to 10.

Common logarithms

This is the way to find the value of log_{2} 70

using a scientific calculator:

find log 70 / log 2.

(Use the 'log' button to write the common log.)

To use log 7 condition,

change log 70 to log 7⋅10.

log 2 = 0.301, log 7 = 0.845

(given conditions)

log 10 = 1

1845/301 = 6.129... .

The given significant numbers are 301, 845:

their digits are 3.

So round 6.129... to the nearest hundreadths: 6.13.

## Example 2

log_{2} 3 is a base 2 log.

So change the base to 2.

Change 18 and 12 to their prime factorizations.

log_{2} 3 = *a*

(given condition)

log_{2} 2 = 1

## Example 3

Change the base to 2.

The given numbers (2, 27, 9, and 16)

either have 2 or 3 as a factor.

So change the base to either 2 or 3.

Change 27, 16, and 9

to their prime factorizations.

Cancel log_{2} 3. (gray)