Explain how to solve 3^x^-4 = 6 using the change of base formula . Include the solution for x in your answer. Round your answer to the nearest thousandth.




To solve the equation \( 3^{x-4} = 6 \) using the change of base formula, follow these steps:

### Step 1: Apply the logarithm to both sides of the equation
Take the logarithm of both sides. You can use any logarithm base, but it's common to use the natural logarithm (ln) or the common logarithm (log). Here, we'll use the natural logarithm:

\[
\ln(3^{x-4}) = \ln(6)
\]

### Step 2: Use the power rule for logarithms
The power rule for logarithms states that \(\ln(a^b) = b \cdot \ln(a)\). Apply this rule to the left side of the equation:

\[
(x-4) \cdot \ln(3) = \ln(6)
\]

### Step 3: Solve for \( x \)
Now, isolate \( x \) by following these steps:

1. Divide both sides by \(\ln(3)\) to get:

\[
x - 4 = \frac{\ln(6)}{\ln(3)}
\]

2. Add 4 to both sides to solve for \( x \):

\[
x = \frac{\ln(6)}{\ln(3)} + 4
\]

### Step 4: Calculate the value of \( x \)
Now, compute the value using a calculator:

\[
x = \frac{\ln(6)}{\ln(3)} + 4 \approx \frac{1.7918}{1.0986} + 4 \approx 1.6309 + 4 \approx 5.631
\]

### Final Answer:
The solution for \( x \) is approximately \( \boxed{5.631} \).

### Do you want details or have any questions?

### Related Questions:
1. What is the change of base formula and why is it useful?
2. How would the solution differ if the equation was \( 3^{x+4} = 6 \)?
3. What if the base was not 3 but another number, like 5? How would the steps change?
4. Could you solve \( 3^{x-4} = 6 \) using logarithms of base 10?
5. How does the power rule for logarithms work in general?
6. How can you check if your solution for \( x \) is correct?
7. What is the difference between natural logarithms and common logarithms?
8. Can you derive the change of base formula from first principles?

### Tip:
Always remember that logarithms can simplify many exponential equations, especially when dealing with unknown exponents.

Learn how to solve the equation 3^(x-4) = 6 using the change of base formula. Detailed step-by-step solution provided, including the application of logarithms and the power rule. Suitable for advanced high school students.

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