https://file.aiquickdraw.com/mathbot/file/1729528141329gibt9i0x.jpg

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!

What is the approximate solution to this equation: 15 * (3)^(2x) = 90?

Learn how to solve the exponential equation 15 * (3)^(2x) = 90 by applying logarithms. This step-by-step solution uses the logarithmic power rule to isolate and solve for x. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation