Question

Use an over/under table to estimate the solution to the exponential equation 3+2^3x =9
to two decimal places.(1 point)
The solution to the exponential equation is x≈

Answer 1

To solve the equation \( 3 + 2^{3x} = 9 \), we can start by isolating the exponential term:

\[
2^{3x} = 9 - 3
\]

\[
2^{3x} = 6
\]

Next, we take the logarithm of both sides. Using base 2 logarithm (although any logarithm base could be used), we have:

\[
3x = \log_2(6)
\]

Now, we can solve for \( x \):

\[
x = \frac{\log_2(6)}{3}
\]

To approximate \( \log_2(6) \), we can use its relationship with logarithms in base 10:

\[
\log_2(6) = \frac{\log_{10}(6)}{\log_{10}(2)}
\]

Calculating \( \log_{10}(6) \) and \( \log_{10}(2) \) using a calculator gives approximately:

\[
\log_{10}(6) \approx 0.7781
\]

\[
\log_{10}(2) \approx 0.3010
\]

So,

\[
\log_2(6) \approx \frac{0.7781}{0.3010} \approx 2.585
\]

Now substitute this back into our expression for \( x \):

\[
x \approx \frac{2.585}{3} \approx 0.86167
\]

Rounding to two decimal places, we find:

\[
x \approx 0.86
\]

Thus, the solution to the exponential equation is:

\[
\boxed{0.86}
\]