To solve the equation [tex]\(\sqrt[3]{277^x} = 729\)[/tex], we'll go through a series of algebraic steps and use logarithms to isolate the variable [tex]\(x\)[/tex]. Here's the detailed step-by-step solution:
1. Start with the given equation:
[tex]\[
\sqrt[3]{277^x} = 729
\][/tex]
2. Rewrite the cube root in exponential form:
[tex]\[
(277^x)^{1/3} = 729
\][/tex]
3. Simplify the left side using exponent rules:
[tex]\[
277^{x/3} = 729
\][/tex]
4. Take the natural logarithm (ln) of both sides to bring the exponent down:
[tex]\[
\ln(277^{x/3}) = \ln(729)
\][/tex]
5. Use the logarithm power rule [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[
\frac{x}{3} \cdot \ln(277) = \ln(729)
\][/tex]
6. Isolate [tex]\(x/3\)[/tex] by dividing both sides by [tex]\(\ln(277)\)[/tex]:
[tex]\[
\frac{x}{3} = \frac{\ln(729)}{\ln(277)}
\][/tex]
7. Solve for [tex]\(x\)[/tex] by multiplying both sides by 3:
[tex]\[
x = 3 \cdot \frac{\ln(729)}{\ln(277)}
\][/tex]
8. Substitute the values found for [tex]\(\ln(729)\)[/tex] and [tex]\(\ln(277)\)[/tex]:
[tex]\[
\ln(729) \approx 6.591673732008658
\][/tex]
[tex]\[
\ln(277) \approx 5.6240175061873385
\][/tex]
9. Calculate the value of [tex]\(x\)[/tex]:
[tex]\[
x = 3 \cdot \frac{6.591673732008658}{5.6240175061873385} \approx 3.5161734781711154
\][/tex]
Thus, the solution to the equation [tex]\(\sqrt[3]{277^x} = 729\)[/tex] is:
[tex]\[
x \approx 3.5161734781711154
\][/tex]
