https://file.aiquickdraw.com/mathbot/file/17262703662510rnphog3.jpg

The equation in the image is:

\[
300 \cdot 2^{2t} = 30,000
\]

Let’s solve this step by step.

### Step 1: Isolate the exponential term
Divide both sides of the equation by 300:

\[
2^{2t} = \frac{30,000}{300} = 100
\]

### Step 2: Solve for \( t \) using logarithms
We now have:

\[
2^{2t} = 100
\]

Take the natural logarithm (or log base 2) of both sides:

\[
\ln(2^{2t}) = \ln(100)
\]

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

\[
2t \cdot \ln(2) = \ln(100)
\]

Now solve for \( t \):

\[
2t = \frac{\ln(100)}{\ln(2)}
\]

\[
t = \frac{\ln(100)}{2 \cdot \ln(2)}
\]

### Step 3: Calculate the value of \( t \)

Let me compute the result.The value of \( t \) is approximately \( 3.32 \).

Let me know if you'd like more details or further explanations!

Here are five related questions:
1. How would you solve the equation if the base were different from 2?
2. What does this equation represent in a real-world context?
3. Can you express the solution using logarithms of other bases?
4. How would the solution change if the number on the right side was different, such as 50,000?
5. What is the significance of using logarithms to solve exponential equations?

**Tip**: Always check if you can simplify the base of the exponent to make solving logarithmic equations easier.

Learn how to solve the exponential equation 300 * 2^(2t) = 30,000 using logarithms. Follow a detailed step-by-step solution, covering key mathematical concepts like exponential equations and logarithmic properties. Suitable for students in Grades 10-12.

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