Use logarithms to solve the given equation. (Round the answer to four decimal places.) 

6(1.095*x* + 1) = 10

*x* = 

We are tasked with solving the equation:

\[
6(1.095^x + 1) = 10
\]

### Step-by-step Solution:

1. **Isolate the exponential term**:
   \[
   6(1.095^x + 1) = 10
   \]
   Divide both sides by 6:
   \[
   1.095^x + 1 = \frac{10}{6}
   \]
   Simplify:
   \[
   1.095^x + 1 = 1.6667
   \]
   Now, subtract 1 from both sides:
   \[
   1.095^x = 0.6667
   \]

2. **Apply logarithms**:
   To solve for \(x\), we take the natural logarithm (or base 10 logarithm) of both sides. I'll use the natural logarithm (\(\ln\)):

   \[
   \ln(1.095^x) = \ln(0.6667)
   \]
   Using the logarithmic property \(\ln(a^b) = b \cdot \ln(a)\), we simplify the left-hand side:
   \[
   x \cdot \ln(1.095) = \ln(0.6667)
   \]

3. **Solve for \(x\)**:
   Now, divide both sides by \(\ln(1.095)\) to isolate \(x\):
   \[
   x = \frac{\ln(0.6667)}{\ln(1.095)}
   \]

   Calculate the values of the logarithms:
   \[
   \ln(0.6667) \approx -0.4055 \quad \text{and} \quad \ln(1.095) \approx 0.0909
   \]
   So:
   \[
   x = \frac{-0.4055}{0.0909} \approx -4.4593
   \]

Thus, the solution is:
\[
x \approx -4.4593
\]

### Do you want further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you apply logarithms to solve equations involving exponents?
2. Why is taking the logarithm of both sides a useful step when solving exponential equations?
3. Can this approach be applied to any base of an exponent?
4. How do we handle exponential equations with different bases on both sides?
5. What other methods could be used to solve exponential equations?

**Tip**: Always remember that you can use either natural logarithms (\(\ln\)) or base-10 logarithms (\(\log\)) when solving exponential equations.

Use logarithms to solve the given equation. (Round the answer to four decimal places.) 6(1.095^x + 1) = 10. x =

Learn how to solve the exponential equation 6(1.095^x + 1) = 10 using logarithms with a step-by-step solution. This guide covers key algebraic concepts such as applying logarithms to isolate variables and solve exponential equations. Suitable for students in Grades 10-12.

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