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
ar
Note that we can use any base in place of a.
Examples on logarithms change of base
Example 1 : Find log
93 .
Solution: Consider the common base as 3.
log
93 = (log
33)/(log
39)
= (log
33) / (log
3(3)
2)
= (log
33) / 2 log
3 3
= 1/2.
Example 2 : Given log
216 = 4. Find log
162 .
Solution: 
Here we have , b = 16, r = 2 and a = 2
log
162 = (log
22)/(log
216)
= (log
22) / (log
2(2)
4)
= (log
22) / 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
23 = x
log
23 = (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
101000 = x
log
101000 = (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
11th grade math
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