What is the value of \log_3 16 \times \log_2 27? — JAMB UTME

What is the value of $\log_3 16 \times \log_2 27$ ?

The value of

\log_3 16 \times \log_2 27

12

By applying the power law of logarithms,

\log_3 16 = 4\log_3 2

and

\log_2 27 = 3\log_2 3

. Multiplying these gives

(4 \times 3)(\log_3 2 \times \log_2 3)

. Using the change of base identity

\log_3 2 \times \log_2 3 = 1

, the product simplifies to

12

Rewrite

16

and

27

as prime powers

\log_3 16 = \log_3 (2^4)

and

\log_2 27 = \log_2 (3^3)

Express numbers in terms of base prime factors to simplify logarithmic powers.

Apply the power law of logarithms

\log_b (a^k) = k \log_b a

\log_3 (2^4) = 4 \log_3 2

and

\log_2 (3^3) = 3 \log_2 3

Bring the exponents out as multipliers.

Multiply the expressions and apply the change of base reciprocal property

\log_a b \times \log_b a = 1

(4 \log_3 2) \times (3 \log_2 3) = 12 \times (\log_3 2 \times \log_2 3) = 12 \times 1 = 12

The logarithmic terms are reciprocals of each other, simplifying their product to 1.

Change of Base and Power Laws of Logarithms

Estimated Time:45s

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