# Logarithms change of base

In this section ask-math explains you the logarithms change of base formula.Any calculator gives us the value of logarithm either base 10 or base 'e'. But if the common and natural logarithm has base other than 10 or 'e' then we use logarithms change of base formula.

**Change of base :**Let us now learn how to convert from base 'b' to any other base 'a' by proving that for any positive real numbers 'r' and 'b', b ≠ 1.

**Proof :**Let N = log

_{b}r

b

^{N}= r ( By definition of logarithm)

Taking log to the base a on both sides, we get

∴ N log

_{a}b = log

_{a}r

Note that we can use any base in place of a.

## Examples on logarithms change of base

**Example 1 :**Find log

_{9}3 .

**Solution:**Consider the common base as 3.

log

_{9}3 = (log

_{3}3)/(log

_{3}9)

= (log

_{3}3) / (log

_{3}(3)

^{2})

= (log

_{3}3) / 2 log

_{3}3

= 1/2.

**Example 2 :**Given log

_{2}16 = 4. Find log

_{16}2 .

**Solution:**

Here we have , b = 16, r = 2 and a = 2

log

_{16}2 = (log

_{2}2)/(log

_{2}16)

= (log

_{2}2) / (log

_{2}(2)

^{4})

= (log

_{2}2) / 4 log

_{2}2

= 1/4.

**Solve each equation using logarithms. Round to the nearest ten-thousandth.**

1) 2

^{x}= 3

**Solution :**2

^{x}= 3

log

_{2}3 = x

log

_{2}3 = (log 3)/(log 2) (consider the base as 10, if not mentioned)

= 0.4771 / 0.3010 (using calculator)

∴ x = 1.5850

2) 8 + 10

^{x}= 1008

**Solution :**

8 + 10

^{x}= 1008

10

^{x}= 1008 - 8

10

^{x}= 1000

log

_{10}1000 = x

log

_{10}1000 = (log 1000)/(log 10)

= (log 10

^{3})/(log 10)

= 3 log 10/ log 10

∴ x = 3

**Use the change of base formula to evaluate each.**

1) log

_{2}9 = log 9/ log 2 = 0.9542/0.3010 = 3.17

2) log

_{4}8 = log 8/ log 4 = 0.9031/0.6021 = 1.499 = 1.5

3) log

_{3}50 = log 50/ log 3 = 1.6989/.4771 = 3.5608

4) log

_{4.6}12.5 = log 12.5/ log 4.6 = 1.0969/0.6627 = 1.655

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