If b is a positive real number unequal to 1 such that \log_b 3 = x and \log_b 5 = y, what … — ACT

If $b$ is a positive real number unequal to $1$ such that $\log_b 3 = x$ and $\log_b 5 = y$ , what is the value of $\log_b \left( \frac{45}{b^2} \right)$ in terms of $x$ and $y$ ?

$2x + y - 2$ Answer
B
$x^2 + y - 2$
C
$\frac{2x + y}{2}$
D
$2x + y - 2b$
E
$2x - y - 2$

Answer

2x + y - 2

The correct answer is

2x + y - 2

. By applying the quotient property of logarithms,

\log_b \left( \frac{45}{b^2} \right)

is rewritten as

\log_b 45 - \log_b(b^2)

. Factoring

45

3^2 \cdot 5

allows the first term to be expanded using the product and power properties into

2\log_b 3 + \log_b 5

. Simplifying

\log_b(b^2)

2

and substituting

x

and

y

yields

2x + y - 2

Step-by-Step Solution

Apply the quotient property of logarithms.

\log_b \left( \frac{45}{b^2} \right) = \log_b 45 - \log_b(b^2)

The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator:

\log_b(M/N) = \log_b M - \log_b N

Factor the number 45 and apply the product property of logarithms.

\log_b(3^2 \cdot 5) - \log_b(b^2) = \log_b(3^2) + \log_b 5 - \log_b(b^2)

Since

45 = 9 \cdot 5 = 3^2 \cdot 5

, we can use the product property:

\log_b(M \cdot N) = \log_b M + \log_b N

Apply the power property to simplify the terms.

2\log_b 3 + \log_b 5 - 2

The power property states that

\log_b(M^k) = k\log_b M

. Also,

\log_b(b^2) = 2

because the base

b

raised to the second power is

b^2

Substitute the given values

x = \log_b 3

and

y = \log_b 5

2x + y - 2

Replacing the logarithmic expressions with

x

and

y

gives the final simplified expression.

Key Concept

Applying logarithmic properties (quotient, product, power) to simplify expressions

Estimated Time:1m 0s

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