To solve the exponential equation [tex]\( 2 \cdot 8^{x-3} = 7 \)[/tex], we need to follow these steps:
### Step-by-Step Solution:
1. Isolate the exponential term:
[tex]\[
8^{x-3} = \frac{7}{2}
\][/tex]
2. Take the logarithm of both sides:
[tex]\[
\log\left(8^{x-3}\right) = \log\left(\frac{7}{2}\right)
\][/tex]
3. Use the power rule of logarithms: The power rule states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex].
[tex]\[
(x-3) \cdot \log(8) = \log\left(\frac{7}{2}\right)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x - 3 = \frac{\log\left(\frac{7}{2}\right)}{\log(8)}
\][/tex]
[tex]\[
x = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} + 3
\][/tex]
5. Determine the exact solution:
[tex]\[
x = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} + 3
\][/tex]
6. Approximate [tex]\(\frac{\log\left(\frac{7}{2}\right)}{\log(8)}\)[/tex] to 4 decimal places:
[tex]\[
\frac{\log\left(3.5\right)}{\log(8)} \approx 0.6024
\][/tex]
Adding 3:
[tex]\[
x \approx 0.6024 + 3 = 3.6024
\][/tex]
### Final Answers:
1. Exact solution:
[tex]\[
x = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} + 3
\][/tex]
2. Approximation to 4 decimal places:
[tex]\[
x \approx 3.6024
\][/tex]
