Solve the exponential equation.
4 Superscript x divided by 3 Baseline equals 7
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
xequals

enter your response here
​(Round to four decimal places as needed. Use a comma to separate answers as​ needed.)
B.
There is no solution

To solve the exponential equation \( \frac{4^x}{3} = 7 \), follow these steps:

### Step 1: Isolate the exponential term
Multiply both sides of the equation by 3 to eliminate the denominator:
\[
4^x = 21
\]

### Step 2: Take the natural logarithm of both sides
Apply the natural logarithm (ln) to both sides of the equation to bring the exponent down:
\[
\ln(4^x) = \ln(21)
\]

Using the logarithmic property \( \ln(a^b) = b \ln(a) \), the equation becomes:
\[
x \ln(4) = \ln(21)
\]

### Step 3: Solve for \( x \)
Divide both sides by \( \ln(4) \):
\[
x = \frac{\ln(21)}{\ln(4)}
\]

### Step 4: Compute the value of \( x \)
Using a calculator, we find:
\[
x = \frac{\ln(21)}{\ln(4)} \approx \frac{3.0445}{1.3863} \approx 2.1968
\]

Thus, the solution is:
\[
x \approx 2.1968
\]

### Final Answer:
The correct choice is **A**: \( x \approx 2.1968 \) (rounded to four decimal places).

---

Would you like more details on any step? Here are some related questions to practice:

1. Solve \( 3^x = 15 \).
2. Solve \( \frac{2^x}{5} = 8 \).
3. Solve \( 5^{x+1} = 125 \).
4. Solve \( 6^x = 200 \) using logarithms.
5. Solve \( \frac{7^x}{2} = 10 \).

**Tip**: Remember that logarithms are useful when solving exponential equations where the variable is in the exponent!

Solve the exponential equation. \( \frac{4^x}{3} = 7 \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x = \) (Round to four decimal places as needed.) B. There is no solution.

Learn how to solve the exponential equation \( \frac{4^x}{3} = 7 \) using logarithms. This step-by-step guide covers isolating the exponential term, applying natural logarithms, and solving for \( x \), with the final answer rounded to four decimal places. Suitable for high school students in Grades 10-12.

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