To solve the equation [tex]\(\log_{\frac{3}{4}} 25 = 3x - 1\)[/tex] for the value of [tex]\(x\)[/tex], follow these steps:
1. Understand the logarithmic equation:
[tex]\[
\log_{\frac{3}{4}} 25 = 3x - 1
\][/tex]
2. Calculate the logarithmic value:
We need to find the value of [tex]\(\log_{\frac{3}{4}} 25\)[/tex]. The logarithm [tex]\(\log_{\frac{3}{4}} 25\)[/tex] represents the power to which the base [tex]\(\frac{3}{4}\)[/tex] must be raised to get 25. The exact calculation will yield:
[tex]\[
\log_{\frac{3}{4}} 25 \approx -11.189
\][/tex]
3. Form the equation:
We now have:
[tex]\[
-11.189 = 3x - 1
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to isolate [tex]\(x\)[/tex]:
[tex]\[
-11.189 + 1 = 3x
\][/tex]
[tex]\[
-10.189 = 3x
\][/tex]
[tex]\[
x = \frac{-10.189}{3}
\][/tex]
[tex]\[
x \approx -3.396
\][/tex]
5. Select the closest match from the given options:
From the list of options provided:
[tex]\[
-3.396, -0.708, 0.304, 0.955
\][/tex]
We see that the approximate value of [tex]\(x\)[/tex] calculated is [tex]\(-3.396\)[/tex], which matches the first option.
Therefore, the approximate value of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{-3.396}
\][/tex]
