Let's solve the equation:

\[
500 = 20 \times (1.5)^{4x}
\]

### Step 1: Divide both sides by 20 to simplify the equation.

\[
\frac{500}{20} = (1.5)^{4x}
\]

This simplifies to:

\[
25 = (1.5)^{4x}
\]

### Step 2: Take the natural logarithm (ln) of both sides to solve for \( x \).

\[
\ln(25) = \ln((1.5)^{4x})
\]

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

\[
\ln(25) = 4x \cdot \ln(1.5)
\]

### Step 3: Solve for \( x \).

Now, divide both sides by \( 4 \cdot \ln(1.5) \):

\[
x = \frac{\ln(25)}{4 \cdot \ln(1.5)}
\]

Now let's calculate the value of \( x \).The solution to the equation is approximately:

\[
x \approx 1.98
\]

Would you like further details on any of the steps or clarifications?

Here are 5 related questions you might find interesting:
1. How do logarithms help in solving exponential equations?
2. What is the logarithmic property \( \ln(a^b) = b \cdot \ln(a) \) used for?
3. Can this equation be solved graphically?
4. How does the base of the exponential affect the solution?
5. What if the base were a different number, like \( 2 \) or \( 3 \)?

**Tip:** Always isolate the exponential term before applying logarithms for easier calculation!

Learn how to solve the exponential equation 500 = 20 * (1.5)^{4x} using logarithms. This step-by-step guide explains key concepts such as dividing both sides, applying natural logarithms, and solving for x. Ideal for high school students in Grades 10-12.

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