Let's solve the equation:
[tex]\[2^{5x} = 13\][/tex]
### Step-by-Step Solution:
1. Take the natural logarithm of both sides:
[tex]\[\ln(2^{5x}) = \ln(13)\][/tex]
2. Apply the logarithmic property [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[5x \cdot \ln(2) = \ln(13)\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[x = \frac{\ln(13)}{5 \cdot \ln(2)}\][/tex]
### Exact Solution:
The exact solution is:
[tex]\[ x = \frac{\ln(13)}{5 \cdot \ln(2)} \][/tex]
### Approximate Solution:
To find the approximate value of [tex]\(x\)[/tex], use the natural logarithms of 13 and 2:
- [tex]\(\ln(13) \approx 2.5649\)[/tex]
- [tex]\(\ln(2) \approx 0.6931\)[/tex]
Now, compute:
[tex]\[ x \approx \frac{2.5649}{5 \cdot 0.6931} \approx 0.7401 \][/tex]
Therefore, the approximate solution to 4 decimal places is:
[tex]\[ x \approx 0.7401 \][/tex]
