https://file.aiquickdraw.com/mathbot/file/1729106855100394j4iz9.jpg

The problem shown in the image is an exponential equation that you are asked to solve algebraically. The equation is:

\[
6^x + 9 = 25
\]

### Step-by-step solution:

1. **Isolate the exponential term:**
   \[
   6^x = 25 - 9
   \]
   Simplifying the right-hand side:
   \[
   6^x = 16
   \]

2. **Apply the logarithm to both sides:**
   To solve for \(x\), apply the natural logarithm (\(\ln\)) to both sides of the equation:
   \[
   \ln(6^x) = \ln(16)
   \]
   Using the logarithmic property \(\ln(a^b) = b \cdot \ln(a)\), the left-hand side becomes:
   \[
   x \cdot \ln(6) = \ln(16)
   \]

3. **Solve for \(x\):**
   Divide both sides by \(\ln(6)\):
   \[
   x = \frac{\ln(16)}{\ln(6)}
   \]

4. **Calculate the value of \(x\):**
   Using approximate values for natural logarithms:
   \[
   \ln(16) \approx 2.7726, \quad \ln(6) \approx 1.7918
   \]
   Therefore:
   \[
   x \approx \frac{2.7726}{1.7918} \approx 1.548
   \]

Thus, the solution is:
\[
x \approx 1.548
\]

Would you like more details or have any questions about this? Here are a few related questions for deeper understanding:

1. How can you solve exponential equations using different logarithmic bases?
2. What is the difference between natural logarithms and common logarithms?
3. How do properties of logarithms simplify solving exponential equations?
4. Can you express the result without logarithms by approximating \(6^x\) iteratively?
5. What happens if the equation were \(6^x + 9 = 30\) instead?

**Tip:** When solving exponential equations, isolating the exponential part and applying logarithms are key techniques.

Solve the exponential equation: 6^x + 9 = 25.

Learn how to solve the exponential equation 6^x + 9 = 25 step-by-step using logarithmic properties. This problem is suitable for students in Grades 9-12, covering key concepts like logarithms and exponential equations.

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