# Change of Base Formula

How to use the change of base formula to simplify the given logarithm: formula, 3 examples, and their solutions.

## Formula

### Formula

log_{a} x = [log_{b} x]/[log_{b} a]

Split a and x

and change the bases to b.

## Example 1

### Example

### Solution

The values of log 2 and log 7 are given.

So change the base to 10.

So log_{2} 70 = [log 70]/[log 2].

Common Logarithm

70 = 7⋅10

log 7⋅10 = log 7 + log 10

Logarithm of a Product

It says

assume log 2 = 0.301, log 7 = 0.845.

So

[log 7 + log 10]/[log 2]

= [0.845 + 1]/[0.301].

0.845 + 1 = 1.845

1.845/0.301 = 1845/301

1845/301 = 6.129...

Round this to the nearest hundreadth.

(0.301, 0.845 has 3 significant digits.

So round 6.129... to make 3 significant digits.)

Then 1845/301 = 6.13.

So 6.13 is the answer.

## Example 2

### Example

### Solution

log_{2} 3 = a is given.

So change the base to 2.

log_{12} 18 = [log_{2} 18]/[log_{2} 12]

Write the prime factorizations of 18 and 12.

18 = 2⋅3^{2}

12 = 2^{2}⋅3

log_{2} (2⋅3^{2}) = log_{2} 2 + log_{2} 3^{2}

log_{2} (2^{2}⋅3) = log_{2} 2^{2} + log_{2} 3

+log_{2} 3^{2} = +2 log_{2} 3

log_{2} 2^{2} = 2 log_{2} 2

log_{2} 2 = 1

log_{2} 3 = a

1 + 2a = 2a + 1

2⋅1 + a = a + 2

So

[2a + 1]/[a + 2]

is the answer.

## Example 3

### Example

### Solution

Change the bases of the logs to 2.

You don't have to change log_{2} 27.

log_{9} 16 = [log_{2} 16]/[log_{2} 9].

You can also change the bases to 3.

You'll get the same answer.

27 = 3^{3}

16 = 2^{4}

9 = 3^{2}

Power

log_{2} 3^{3} = 3 log_{2} 3

log_{2} 2^{4} = 4 log_{2} 2

log_{2} 3^{2} = 2 log_{2} 3

Cancel (log_{2} 3).

log_{2} 2 = 1

4⋅1/2 = 2

3⋅2 = 6

So 6 is the answer.