Solve the exponential equation. Write the exact answer with natural logarithms and then approximate the result correct to three decimal places.
2 Superscript 5 x plus 3 Baseline equals 15
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your choice.
A.
The exact​ answer(s) with natural logarithms​ is/are x equals nothing

.
​(Type an exact answer in simplified form. Type an integer or a fraction. Use a comma to separate answers as​ needed.)
The approximate​ answer(s) is/are x almost equals nothing

.
​(Simplify your answer. Type an integer or a decimal. Round to three decimal places as needed. Use a comma to separate answers as​ needed.)
B.
There is no solution. The solution set is the empty​ set, empty set.

We are given the equation:

\[
2^{5x + 3} = 15
\]

### Step 1: Isolate the exponential term
First, divide both sides of the equation by 2³ to remove the constant on the left-hand side:

\[
2^{5x} = \frac{15}{2^3} = \frac{15}{8}
\]

### Step 2: Take the natural logarithm of both sides
To solve for \(x\), take the natural logarithm (ln) of both sides:

\[
\ln(2^{5x}) = \ln\left(\frac{15}{8}\right)
\]

Using the logarithmic property \(\ln(a^b) = b\ln(a)\), we simplify:

\[
5x \cdot \ln(2) = \ln\left(\frac{15}{8}\right)
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\):

\[
x = \frac{\ln\left(\frac{15}{8}\right)}{5 \ln(2)}
\]

This is the exact solution in terms of natural logarithms.

### Step 4: Approximate the result
To approximate the result, use the values \(\ln(2) \approx 0.6931\) and \(\ln\left(\frac{15}{8}\right) \approx 0.6213\):

\[
x \approx \frac{0.6213}{5 \times 0.6931} \approx \frac{0.6213}{3.4655} \approx 0.1793
\]

Rounding to three decimal places, we get:

\[
x \approx 0.179
\]

### Final Answer:
- **Exact solution:** \(x = \frac{\ln\left(\frac{15}{8}\right)}{5 \ln(2)}\)
- **Approximate solution:** \(x \approx 0.179\)

Would you like more details on logarithmic properties or solving exponential equations?

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Here are 5 related questions:
1. How do you solve equations with logarithms and exponents together?
2. What is the relationship between logarithms and exponentials?
3. How can natural logarithms be used to solve real-world problems?
4. How do you simplify expressions involving logarithms and powers?
5. What are the key properties of logarithms that help solve equations?

**Tip:** When solving logarithmic or exponential equations, always remember the key properties of logarithms to simplify your expressions efficiently!

Solve the exponential equation. Write the exact answer with natural logarithms and then approximate the result correct to three decimal places. 2^(5x + 3) = 15.

Learn how to solve the exponential equation 2^(5x + 3) = 15 using natural logarithms. This problem demonstrates key concepts in algebra and logarithms, showing exact and approximate solutions. Suitable for high school students in Grades 10-12.

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