Given that \log_a b = 2 and \log_b c = 3, what is the value of \log_{a b c} (a^3 b^2 c)? — JAMB UTME

Given that $\log_a b = 2$ and $\log_b c = 3$ , what is the value of $\log_{a b c} (a^3 b^2 c)$ ?

\frac{13}{9}

Using the change of base chain rule,

\log_a c = \log_a b \cdot \log_b c = 2 \times 3 = 6

. Changing the base of the target expression to base

a

gives

\frac{\log_a (a^3 b^2 c)}{\log_a (a b c)}

. Expanding both terms using product and power rules gives numerator

3(1) + 2(2) + 6 = 13

and denominator

1 + 2 + 6 = 9

, resulting in

\frac{13}{9}

Express

\log_a c

using the change of base relationship.

\log_a c = \log_a b \cdot \log_b c = 2 \times 3 = 6

By the change of base rule (chain rule of logarithms),

\log_a b \cdot \log_b c = \log_a c

Apply the change of base formula to convert

\log_{a b c} (a^3 b^2 c)

to base

a

\log_{a b c} (a^3 b^2 c) = \frac{\log_a (a^3 b^2 c)}{\log_a (a b c)}

The change of base formula states that

\log_B X = \frac{\log_a X}{\log_a B}

Expand the numerator and denominator using logarithmic product and power rules.

Numerator:

3\log_a a + 2\log_a b + \log_a c = 3(1) + 2(2) + 6 = 13

. Denominator:

\log_a a + \log_a b + \log_a c = 1 + 2 + 6 = 9

\log_a (X Y Z) = \log_a X + \log_a Y + \log_a Z

and

\log_a (X^k) = k \log_a X

Divide the expanded numerator by the denominator.

\frac{13}{9}

Substituting the computed values yields the simplified fraction

\frac{13}{9}

Change of Base Rule and Logarithmic Expansion Laws

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