The equation shown in your image is:

$15 \cdot 3^{2x} = 90$

To solve for $x$, we can follow these steps:

Step 1: Isolate the exponential term

Divide both sides of the equation by 15:

$3^{2x} = \frac{90}{15} = 6$

Step 2: Take the logarithm of both sides

We can now apply the natural logarithm (or log base 10) to both sides to deal with the exponent:

$\log(3^{2x}) = \log(6)$

Step 3: Use the power rule of logarithms

The logarithmic power rule states that $\log(a^b) = b \log(a)$. Applying this rule:

$2x \log(3) = \log(6)$

Step 4: Solve for $x$

Now, solve for $x$:

$x = \frac{\log(6)}{2 \log(3)}$

This matches the option:

$\boxed{x = \frac{\log 6}{2 \log 3}}$

Final Answer:

The correct answer is the second option: $x = \frac{\log 6}{2 \log 3}$

Would you like a deeper explanation or have any follow-up questions? Here are some related questions to explore further:

1. What are logarithmic properties and how do they help in solving exponential equations?
2. How would you solve if the equation was $15(3^{2x+1}) = 90$?
3. What happens if you use a different logarithmic base, like base 2?
4. How do we use logarithms in real-world problems, like compound interest?
5. Can you solve the equation $2^x = 5$ using logarithms?

Tip: The power rule of logarithms is very useful when dealing with exponents. Always remember to apply it when you have a variable in the exponent!