To solve the equation $3^{2x+3} = 15$ for $x$, follow these steps:

Step 1: Take the logarithm of both sides

Since the equation involves an exponential function, the natural logarithm (ln) is a good choice to help solve for $x$:

$\ln(3^{2x+3}) = \ln(15)$

Step 2: Apply the logarithm power rule

The logarithm power rule states that $\ln(a^b) = b\ln(a)$. Applying this rule to the left side:

$(2x + 3) \ln(3) = \ln(15)$

Step 3: Isolate the term containing $x$

Next, solve for $x$ by isolating the term involving $x$:

$2x + 3 = \frac{\ln(15)}{\ln(3)}$

Step 4: Solve for $x$

Now, isolate $x$ by first subtracting 3 from both sides and then dividing by 2:

$2x = \frac{\ln(15)}{\ln(3)} - 3$

$x = \frac{1}{2} \left(\frac{\ln(15)}{\ln(3)} - 3\right)$

Step 5: Simplify (if needed)

This is the exact form of the solution. However, you can compute the numerical value using logarithms:

$x = \frac{1}{2} \left(\frac{\ln(15)}{\ln(3)} - 3\right)$

Using approximate values of natural logarithms:

$\ln(15) \approx 2.708, \quad \ln(3) \approx 1.099$

Substituting these values:

$x \approx \frac{1}{2} \left(\frac{2.708}{1.099} - 3\right)$

$x \approx \frac{1}{2} \left(2.465 - 3\right)$

$x \approx \frac{1}{2} \times (-0.535) \approx -0.2675$

Final Answer:

$x \approx -0.2675$

Would you like further explanation or have any other questions? Here are five related questions:

1. How would the solution change if the equation were $3^{2x+3} = 81$?
2. What if the base of the exponent was $e$ instead of 3?
3. How can you solve $3^{2x+3} = 15$ using a different logarithm base?
4. What happens if the exponent is a more complex expression like $2x + 3y$?
5. Can this method be used to solve $2^{3x+1} = 10$? How would the steps differ?

Tip: When dealing with exponential equations, taking the logarithm of both sides is a common and effective technique for solving for the unknown exponent.