Answer :
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]
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]
