To find the [tex]\( x \)[/tex]-intercept of the function [tex]\( f(x) = 3 \log (x + 5) + 2 \)[/tex], we need to determine the value of [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex]. Here is the step-by-step process:
1. Set the function equal to 0:
[tex]\[
3 \log (x + 5) + 2 = 0
\][/tex]
2. Isolate the logarithmic term:
Subtract 2 from both sides to isolate the logarithmic term:
[tex]\[
3 \log (x + 5) = -2
\][/tex]
3. Solve for the logarithm:
Divide both sides of the equation by 3 to solve for the logarithm:
[tex]\[
\log (x + 5) = \frac{-2}{3}
\][/tex]
4. Rewrite the logarithmic equation in exponential form:
Recall that [tex]\( \log a = b \)[/tex] means that [tex]\( 10^b = a \)[/tex]. So, we rewrite the equation as:
[tex]\[
x + 5 = 10^{\frac{-2}{3}}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Subtract 5 from both sides:
[tex]\[
x = 10^{\frac{-2}{3}} - 5
\][/tex]
So, the [tex]\( x \)[/tex]-intercept is given by:
[tex]\[ x = 10^{\frac{-2}{3}} - 5 \][/tex]
Now, compare this with the given answer choices:
A. [tex]\( 10^{2/3} - 5 \)[/tex]
B. [tex]\( 10^{2/3} + 5 \)[/tex]
C. [tex]\( 10^{-2/3} - 5 \)[/tex]
D. [tex]\( 10^{-2/3} + 5 \)[/tex]
The correct answer is:
C. [tex]\( 10^{-2/3} - 5 \)[/tex]
