For all positive real numbers x and y, which of the following expressions is equivalent to… — ACT

For all positive real numbers $x$ and $y$ , which of the following expressions is equivalent to $\log_3(9x^4 y^{-2})$ ?

2 + 4\log_3(x) - 2\log_3(y)

The correct answer is correct because applying the product and power properties of logarithms allows us to expand the argument

9x^4 y^{-2}

. Specifically, the expression can be written as

\log_3(9) + \log_3(x^4) + \log_3(y^{-2})

. Since

3^2 = 9

, the term

\log_3(9)

evaluates to

2

. Applying the power rule

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

to the remaining terms yields

4\log_3(x)

and

-2\log_3(y)

. Combining these terms results in the equivalent expression.

Use the product rule of logarithms,

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

, to separate the factors in the argument.

\log_3(9x^4 y^{-2}) = \log_3(9) + \log_3(x^4) + \log_3(y^{-2})

This allows the expression to be broken down into individual terms for each base and variable.

Evaluate the constant logarithm

\log_3(9)

. Since

3^2 = 9

, this simplifies to

2

\log_3(9) = 2

To simplify the numerical term to its integer value.

Use the power rule of logarithms,

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

, to bring the exponents of the variables to the front of each logarithm.

\log_3(x^4) = 4\log_3(x)

and

\log_3(y^{-2}) = -2\log_3(y)

To isolate the variable terms inside simpler logarithmic expressions.

Combine all the simplified parts into a single expression.

2 + 4\log_3(x) - 2\log_3(y)

To write the final expanded equivalent expression.

Logarithmic properties, including product, power, and evaluation of basic log terms.

Substitute test values for the variables. For example, let

x = 3

and

y = 3

. The original expression evaluates to

\log_3(9 \cdot 3^4 \cdot 3^{-2}) = \log_3(81) = 4

. Substituting

x = 3

and

y = 3

into the correct expression yields

2 + 4\log_3(3) - 2\log_3(3) = 2 + 4(1) - 2(1) = 4

, which matches the original expression's value.

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