If \log_4 x = \frac{3}{2}, what is the value of x? — ACT

If $\log_4 x = \frac{3}{2}$ , what is the value of $x$ ?

The correct answer is

8

To solve the equation

\log_4 x = \frac{3}{2}

, we apply the definition of a logarithm to rewrite it in exponential form:

x = 4^{3/2}

. We then evaluate the exponent by first taking the square root of

4

, which is

2

, and then cubing it to get

2^3 = 8

. Thus, the value of

x

8

Rewrite the logarithmic equation in its equivalent exponential form.

x = 4^{3/2}

By definition, a logarithmic equation of the form

\log_b a = c

is equivalent to the exponential equation

b^c = a

Evaluate the exponential expression

4^{3/2}

x = 8

The fractional exponent can be simplified by taking the square root of the base first, which is

\sqrt{4} = 2

, and then raising the result to the power of the numerator, giving

2^3 = 8

Converting logarithmic equations to exponential form and evaluating fractional exponents

You can solve this by substituting the answer choices back into the equation. For example, testing the value

8

gives

\log_4 8 = \log_4 (2^3) = 3 \log_4 2

. Since

2

is the square root of

4

\log_4 2 = \frac{1}{2}

. Therefore,

3 \times \frac{1}{2} = \frac{3}{2}

, confirming that

8

is the correct solution.

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