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!