Answer :

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]