To solve the exponential equation \( 4^{x+7} = 8 \), let's use the properties of logarithms to find \( x \). Here's a detailed step-by-step solution:
1. Rewrite the equation in logarithmic form:
[tex]\[
4^{x+7} = 8
\][/tex]
To solve for \( x \), we will apply the natural logarithm to both sides of the equation because logarithms allow us to bring the exponent down as a coefficient:
[tex]\[
\ln(4^{x+7}) = \ln(8)
\][/tex]
2. Use the property of logarithms:
One of the important properties of logarithms is that \( \ln(a^b) = b \cdot \ln(a) \). Applying this property, we get:
[tex]\[
(x + 7) \cdot \ln(4) = \ln(8)
\][/tex]
3. Isolate \( x \):
To isolate \( x \), we need to divide both sides of the equation by \( \ln(4) \):
[tex]\[
x + 7 = \frac{\ln(8)}{\ln(4)}
\][/tex]
4. Solve for \( x \):
Subtract 7 from both sides to solve for \( x \):
[tex]\[
x = \frac{\ln(8)}{\ln(4)} - 7
\][/tex]
Now, let's express the solution numerically using the values:
- \( \ln(8) \approx 2.0794415416798357 \)
- \( \ln(4) \approx 1.3862943611198906 \)
Using these values, we calculate:
[tex]\[
x = \frac{2.0794415416798357}{1.3862943611198906} - 7
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{2.0794415416798357}{1.3862943611198906} \approx 1.5
\][/tex]
Finally, subtract 7:
[tex]\[
x = 1.5 - 7 = -5.5
\][/tex]
So, the solution to the equation \( 4^{x+7} = 8 \) is:
[tex]\[
x = -5.5
\][/tex]
Expressing the solution using natural logarithms, the answer is given by:
[tex]\[
x = \frac{\ln(8)}{\ln(4)} - 7
\][/tex]
