What is the value of \frac{1}{\log_2 100} + \frac{1}{\log_5 100}? — JAMB UTME

What is the value of $\frac{1}{\log_2 100} + \frac{1}{\log_5 100}$ ?

\frac{1}{2}

Using the change of base identity

\frac{1}{\log_a b} = \log_b a

, the expression simplifies to

\log_{100} 2 + \log_{100} 5

. By the product rule of logarithms, this equals

\log_{100}(2 \times 5) = \log_{100} 10

. Since

100^{\frac{1}{2}} = 10

, the value is

\frac{1}{2}

Apply the reciprocal change of base rule

\frac{1}{\log_a b} = \log_b a

\frac{1}{\log_2 100} = \log_{100} 2

and

\frac{1}{\log_5 100} = \log_{100} 5

Converting to a common base of

100

allows the use of logarithmic addition laws.

Combine the two logarithms using the product law

\log_c x + \log_c y = \log_c (xy)

\log_{100} 2 + \log_{100} 5 = \log_{100} (2 \times 5) = \log_{100} 10.

Adding logarithms with the same base is equivalent to taking the logarithm of the product of their arguments.

Evaluate

\log_{100} 10

Since

100^{\frac{1}{2}} = 10

\log_{100} 10 = \frac{1}{2}

The logarithm asks what power base

100

must be raised to in order to equal

10

Logarithms and Change of Base

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