Change of Base Formula

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)

logb c = (loga c) / (loga b)

a is the base.
So 0 < a < 1, a > 1.

Proof

Change of Base Formula: Proof of the formula

Set b(brown) = c.

Log both sides. (base: b)
(brown) = logb c

Logarithmic form

See b(brown) = c again.

Log both sides. (base: a)
loga b(brown) = loga c

Product of powers

(brown) = (loga c) / (loga b)
And (brown) = logb c.

So logb c = (loga c) / (loga b).

Example 1

Find the value of the given expresssion. (Assume log 2 = 0.301, log 7 = 0.845.) log_2 70

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 log2 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.

Logarithms of products

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

If log_2 3 = a, find the value of the given expression. log_12 18

log2 3 is a base 2 log.

So change the base to 2.

Change 18 and 12 to their prime factorizations.

Logarithms of products

Logarithms of powers

log2 3 = a
(given condition)

log2 2 = 1

Example 3

Simplify the given expression. (log_2 27)(log_9 16)

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.

Logarithms of powers

Cancel log2 3. (gray)