What is the numerical value of the expression \log_2(\log_3 81) + \log_5\left(\frac{1}{125… — JAMB UTME

What is the numerical value of the expression $\log_2(\log_3 81) + \log_5\left(\frac{1}{125}\right) + 4^{\log_2 3}$ ?

Answer: 8

Evaluating each term individually: the nested logarithm

\log_2(\log_3 81) = \log_2 4 = 2

; the reciprocal log argument

\log_5(1/125) = -3

; and the exponential log power

4^{\log_2 3} = (2^{\log_2 3})^2 = 3^2 = 9

. Combining these gives

2 - 3 + 9 = 8

Evaluate the first component

\log_2(\log_3 81)

Since

81 = 3^4

, the inner expression

\log_3 81 = 4

, and subsequently

\log_2 4 = 2

Evaluate the second component

\log_5\left(\frac{1}{125}\right)

-3

Since

\frac{1}{125} = 5^{-3}

, applying the power law of logarithms gives

-3

Evaluate the third component

4^{\log_2 3}

Rewrite

4

2^2

to obtain

(2^{\log_2 3})^2 = 3^2 = 9

using the fundamental identity

a^{\log_a b} = b

Sum the results of the three components

Calculate

2 + (-3) + 9 = 8

Properties of logarithms including change of power, negative exponents, and logarithm exponentiation identities

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