We are tasked with solving the exponential equation:

$2(1.15)^x + 1 = 8$

Step 1: Isolate the exponential term

First, subtract 1 from both sides:

$2(1.15)^x = 7$

Next, divide both sides by 2:

$(1.15)^x = 3.5$

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Step 2: Solve for $x$

Take the natural logarithm ($\ln$) of both sides to bring $x$ out of the exponent:

$\ln((1.15)^x) = \ln(3.5)$

Using the logarithmic property $\ln(a^b) = b\ln(a)$, we rewrite this as:

$x \ln(1.15) = \ln(3.5)$

Solve for $x$:

$x = \frac{\ln(3.5)}{\ln(1.15)}$

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Step 3: Compute the values

Using a calculator:

$\ln(3.5) \approx 1.25276, \quad \ln(1.15) \approx 0.13976$

Substitute these values:

$x = \frac{1.25276}{0.13976} \approx 8.965$

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Final Answer:

The solution is:

$x \approx 8.965$

Choice:

The correct choice is:

$\text{A. The solution set is } \{8.965\}.$

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Would you like further clarification?

Related Questions:

1. What are the general steps to solve any exponential equation?
2. How do logarithmic properties help in solving exponential equations?
3. What is the significance of natural logarithms ($\ln$) in exponential equations?
4. Can you solve $5 \cdot (1.2)^x + 2 = 15$ following the same steps?
5. What happens if the exponential equation has no solution?

Tip:

Always isolate the exponential term before applying logarithms to simplify calculations.