https://file.aiquickdraw.com/mathbot/file/173048363249559jvemld.jpg

The equation presented in the image is:

\[
4^x = 21
\]

To solve for \( x \), we can use logarithms.

### Solution

1. Take the natural logarithm (or common logarithm) of both sides:
   \[
   \ln(4^x) = \ln(21)
   \]

2. Use the power rule for logarithms, which states \( \ln(a^b) = b \ln(a) \):
   \[
   x \cdot \ln(4) = \ln(21)
   \]

3. Isolate \( x \) by dividing both sides by \( \ln(4) \):
   \[
   x = \frac{\ln(21)}{\ln(4)}
   \]

4. Calculate the value using approximate values for the natural logarithms:
   \[
   x \approx \frac{3.0445}{1.3863} \approx 2.195
   \]

### Answer:
\[
x \approx 2.195
\]

Would you like a deeper explanation or have any questions?

### Further Questions to Explore
1. What is the effect of changing the base (4 in this case) on the solution of the equation?
2. How would the solution change if the equation were \( 4^x = 40 \)?
3. Can we solve this equation using a different logarithm base, such as base 10?
4. How do logarithmic functions relate to exponential functions like \( 4^x \)?
5. What are some real-life applications of solving exponential equations like this one?

### Tip:
When solving exponential equations, using logarithms is often the most straightforward method to isolate the variable in the exponent.

Learn how to solve the exponential equation 4^x = 21 using logarithms. This detailed solution explains the steps, including applying the power rule of logarithms to isolate the variable. Ideal for students in Grades 10-12.

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