If a, b, and c are positive real numbers greater than 1 such that \log_b(a) = \frac{3}{2} … — ACT

If $a$ , $b$ , and $c$ are positive real numbers greater than $1$ such that $\log_b(a) = \frac{3}{2}$ and $\log_c(b) = \frac{4}{3}$ , what is the value of $\log_a(abc)$ ?

A
$\frac{1}{3}$
$\frac{13}{6}$ Answer
C
$\frac{23}{9}$
D
$3$
E
$\frac{9}{2}$

Answer

\frac{13}{6}

The correct answer is obtained by expanding

\log_a(abc)

into

\log_a(a) + \log_a(b) + \log_a(c)

. Since

\log_a(a) = 1

and

\log_a(b)

is the reciprocal of

\log_b(a)

, we find

\log_a(b) = \frac{2}{3}

. Using the chain rule of change of base, we calculate

\log_a(c) = \log_a(b) \cdot \log_b(c) = \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}

. Adding these three parts together yields

1 + \frac{2}{3} + \frac{1}{2} = \frac{13}{6}

Step-by-Step Solution

Apply the logarithmic product rule to expand the target expression.

\log_a(abc) = \log_a(a) + \log_a(b) + \log_a(c)

The logarithm of a product is equal to the sum of the logarithms of its individual factors.

Evaluate the first two terms of the expanded expression.

\log_a(a) = 1

, and since

\log_b(a) = \frac{3}{2}

, we have

\log_a(b) = \frac{1}{\log_b(a)} = \frac{2}{3}

The logarithm of a base to itself is always

1

, and the base reciprocal property states that

\log_x(y) = \frac{1}{\log_y(x)}

Evaluate the third term

\log_a(c)

using base properties and the change of base formula.

Since

\log_c(b) = \frac{4}{3}

, we have

\log_b(c) = \frac{3}{4}

. Then,

\log_a(c) = \log_a(b) \cdot \log_b(c) = \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}

By applying the change of base formula, we can rewrite

\log_a(c)

in terms of base

b

\frac{\log_b(c)}{\log_b(a)} = \log_a(b) \cdot \log_b(c)

Sum the three evaluated logarithmic terms together.

1 + \frac{2}{3} + \frac{1}{2} = \frac{6}{6} + \frac{4}{6} + \frac{3}{6} = \frac{13}{6}

Combining the values of the individual terms gives the final value of the expanded expression.

Key Concept

Properties of Logarithms and Change of Base Formula

Estimated Time:2m 0s

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