Before learning the power rule of logarithms, we have to be aware of the parts of a logarithm.
Consider the logarithm given below.
log_{b}a
In the logarithm above, 'a' is called argument and 'b' is called base.
Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.
log_{a}m^{n} = nlog_{a}m
In other words, if there is a value multiplied in front of a logarithm, the value can be taken as exponent to the argument.
alog_{x}b = log_{x}a^{b}
Apart from the power rule of logarithms, there are two other important rules of logarithm.
(i) Product Rule
(ii) Quotient Rule
Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base.
log_{a}mn = log_{a}m + log_{a}n
Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base.
log_{a}(m/n) = log_{a}m - log_{a}n
Problem 1 :
Find the logarithm of 64 to the base 4.
Solution :
Write 64 as a power of 4.
64 = 4 x 4 x 4
= 4^{3}
log_{4}64 = log_{4}(4)^{3}
= 3log_{4}4
= 3(1)
= 3
Problem 2 :
Find the logarithm 1728 to the base 2√3.
Solution :
Write 1728 as a power of 2√3.
1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3
1728 = 2^{6} x 3^{3}
1728 = 2^{6} x [(√3)^{2}]^{3}
1728 = 2^{6} x (√3)^{6}
1728 = (2√3)^{6}
log_{2}_{√3}(1728) = log_{2}_{√3}(2√3)^{6}
Using the power rule of logarithms,
= 6log_{2}_{√3}(2√3)
= 6(1)
= 6
Problem 3 :
Find the logarithm of 0.0001 to the base 0.1.
Solution :
log_{0.1}0.0001 = log_{0.1}(0.1)^{4}
= 4log_{0.1}0.1
= 4(1)
= 4
Problem 4 :
Find the logarithm 1/64 to the base 4.
Solution :
log_{4}(1/64) = log_{4}1 - log_{4}64
= 0 - log_{4}(4)^{3}
= -3log_{4}4
= -3(1)
= -3
Problem 5 :
Find the logarithm of 0.3333...... to the base 3.
Solution :
log_{3}(0.3333......) = log_{3}(1/3)
= log_{3}1 - log_{3}3
= 0 - 1
= -1
Problem 6 :
If log_{y}(√2) = 1/4, find the value of y.
Write the equation in exponential form.
√2 = y^{1/4}
Raise to the power 4 on both sides.
(√2)^{4} = (y^{1/4})^{4}
(2^{1/2})^{4} = y
2^{2} = y
4 = y
Problem 7 :
Simplify :
(1/2)log_{10}25 - 2log_{10}3 + log_{10}18
Solution :
= (1/2)log_{10}25 - 2log_{10}3 + log_{10}18
Using power rule of logarithms,
= log_{10}25^{1/2} - log_{10}3^{2} + log_{10}18
= log_{10}(5^{2})^{1/2} - log_{10}3^{2} + log_{10}18
= log_{10}5 - log_{10}9 + log_{10}18
= log_{10}5 + log_{10}18 - log_{10}9
Using the product rule of logarithms,
= log_{10}(5 x 18) - log_{10}9
= log_{10}90 - log_{10}9
Using the quotient rule of logarithms,
= log_{10}(90/9)
= log_{10}10
= 1
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