Question

Solve log subscript 81 baseline 32 x minus log subscript 81 baseline left parenthesis x minus 3 right parenthesis equals start fraction 3 over 4 end fraction for x.
(1 point)
Responses

x equals start fraction 24 over 31 end fraction
Image with alt text: x equals start fraction 24 over 31 end fraction

x equals start fraction negative 81 over 5 end fraction
Image with alt text: x equals start fraction negative 81 over 5 end fraction

x equals start fraction 729 root index 3 start root 3 end root over 243 root index 3 start root 3 end root minus 32 end fraction
Image with alt text: x equals start fraction 729 root index 3 start root 3 end root over 243 root index 3 start root 3 end root minus 32 end fraction

no solution

Answer 1

To solve the equation

\[
\log_{81}(32x) - \log_{81}(x - 3) = \frac{3}{4},
\]

we can use the properties of logarithms, specifically that the subtraction of logarithms can be rewritten as a logarithm of a quotient:

\[
\log_{81}\left(\frac{32x}{x - 3}\right) = \frac{3}{4}.
\]

We can eliminate the logarithm by exponentiating both sides with base 81:

\[
\frac{32x}{x - 3} = 81^{\frac{3}{4}}.
\]

Calculating \(81^{\frac{3}{4}}\):

Since \(81 = 3^4\), we have:

\[
81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27.
\]

Now we replace \(81^{\frac{3}{4}}\) in our equation:

\[
\frac{32x}{x - 3} = 27.
\]

Next, we can cross-multiply:

\[
32x = 27(x - 3).
\]

Expanding the right side gives:

\[
32x = 27x - 81.
\]

Now, we can isolate \(x\):

\[
32x - 27x = -81,
\]
\[
5x = -81.
\]
\[
x = -\frac{81}{5}.
\]

Thus, the solution for \(x\) is

\[
x = -\frac{81}{5}.
\]

Therefore, the correct answer from the provided responses is:

**x equals start fraction negative 81 over 5 end fraction.**