QuestionSolve the following equation. $\log _{x} 125=-3$
Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \}. (Simplify your answer.) B. The solution set is $\{x \mid x>0, x \neq 1\}$ .

Studdy Solution

STEP 1

1. The equation involves a logarithm with base $x$ .
2. We need to find the value of $x$ that satisfies the equation.
3. The logarithmic equation can be rewritten in exponential form.

STEP 2

1. Rewrite the logarithmic equation in exponential form.
2. Solve the resulting exponential equation for $x$ .
3. Determine the solution set.

STEP 3

Rewrite the logarithmic equation in exponential form. The equation given is:
$\log_{x} 125 = -3$
This can be rewritten as:
$x^{-3} = 125$

STEP 4

Solve the exponential equation for $x$ . We have:
$x^{-3} = 125$
To solve for $x$ , take the reciprocal of both sides to eliminate the negative exponent:
$x^3 = \frac{1}{125}$
Now, take the cube root of both sides:
$x = \sqrt[3]{\frac{1}{125}}$
Since $125 = 5^3$ , we have:
$x = \frac{1}{5}$

STEP 5

Determine the solution set. Since we found $x = \frac{1}{5}$ , the solution set is:
$\boxed{\left\{\frac{1}{5}\right\}}$
The correct answer is: A. The solution set is $\left\{\frac{1}{5}\right\}$ .

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