To solve for [tex]\(x\)[/tex] in the equation [tex]\(10(1.5)^{3x} = 80\)[/tex], follow these steps:
1. Isolate the exponential term:
[tex]\[
(1.5)^{3x} = \frac{80}{10}
\][/tex]
[tex]\[
(1.5)^{3x} = 8
\][/tex]
2. Take the natural logarithm of both sides:
[tex]\[
\ln\left((1.5)^{3x}\right) = \ln(8)
\][/tex]
3. Use the properties of logarithms to bring the exponent down:
[tex]\[
3x \cdot \ln(1.5) = \ln(8)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(8)}{3 \cdot \ln(1.5)}
\][/tex]
Upon calculating the values:
[tex]\[
\ln(8) \approx 2.0794
\][/tex]
[tex]\[
3 \cdot \ln(1.5) \approx 1.2164
\][/tex]
5. Divide and round to the nearest hundredth:
[tex]\[
x = \frac{2.0794}{1.2164} \approx 1.71
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation is:
[tex]\[
\boxed{1.71}
\][/tex]
