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

loga x = [logb x]/[logb 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 log2 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

log2 3 = a is given.

So change the base to 2.

log12 18 = [log2 18]/[log2 12]

Write the prime factorizations of 18 and 12.

18 = 2⋅32
12 = 22⋅3

log2 (2⋅32) = log2 2 + log2 32

log2 (22⋅3) = log2 22 + log2 3

+log2 32 = +2 log2 3

log2 22 = 2 log2 2

log2 2 = 1

log2 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 log2 27.

log9 16 = [log2 16]/[log2 9].

You can also change the bases to 3.
You'll get the same answer.

27 = 33

16 = 24

9 = 32

Power

log2 33 = 3 log2 3

log2 24 = 4 log2 2

log2 32 = 2 log2 3

Cancel (log2 3).

log2 2 = 1

4⋅1/2 = 2

3⋅2 = 6

So 6 is the answer.