To find the [tex]\( x \)[/tex]-intercept of the graph of the function [tex]\( f(x) = 3 \log (x + 5) - 2 \)[/tex], we need to determine the value of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex].
1. Set the function equal to zero:
[tex]\[
0 = 3 \log (x + 5) - 2
\][/tex]
2. Isolate the logarithmic term:
[tex]\[
3 \log (x + 5) = 2
\][/tex]
3. Divide both sides by 3 to further isolate the logarithmic expression:
[tex]\[
\log(x + 5) = \frac{2}{3}
\][/tex]
4. To eliminate the logarithm, we exponentiate both sides. Assuming the base of the logarithm is 10 (common logarithm):
[tex]\[
x + 5 = 10^{\frac{2}{3}}
\][/tex]
5. Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[
x = 10^{\frac{2}{3}} - 5
\][/tex]
Based on this calculation, the [tex]\( x \)[/tex]-intercept of the graph is [tex]\( 10^{\frac{2}{3}} - 5 \)[/tex]. Therefore, the correct answer is:
C. [tex]\( 10^{2 / 3} - 5 \)[/tex]
