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To solve the equation \(3^x - 4 = 6\) using the change of base formula, we will proceed step-by-step:

### Step 1: Isolate the Exponential Expression
Start by isolating the term involving the exponential expression:

\[
3^x - 4 = 6
\]

Add 4 to both sides:

\[
3^x = 10
\]

### Step 2: Apply Logarithm to Both Sides
To solve for \(x\), take the logarithm of both sides. We can use any logarithm, but for consistency, let's use the natural logarithm (base \(e\)):

\[
\ln(3^x) = \ln(10)
\]

### Step 3: Use the Logarithm Power Rule
Apply the power rule of logarithms, which states \(\ln(a^b) = b \ln(a)\):

\[
x \ln(3) = \ln(10)
\]

### Step 4: Solve for \(x\)
Now, solve for \(x\) by dividing both sides by \(\ln(3)\):

\[
x = \frac{\ln(10)}{\ln(3)}
\]

### Step 5: Use the Change of Base Formula
The change of base formula for logarithms states that:

\[
\log_b(y) = \frac{\log_k(y)}{\log_k(b)}
\]

In this case, using natural logarithms (\(\ln\)) or common logarithms (\(\log\)) provides the same solution. Therefore:

\[
x = \frac{\log(10)}{\log(3)}
\]

### Step 6: Calculate the Value of \(x\)
Now, compute the value using a calculator:

\[
x \approx \frac{\log(10)}{\log(3)} \approx \frac{1}{0.4771} \approx 2.096
\]

### Step 7: Round to the Nearest Thousandth
Rounded to the nearest thousandth, the value of \(x\) is:

\[
x \approx 2.096
\]

### Final Answer
The solution for \(x\) is approximately \(2.096\).

Would you like more details or have any questions?

### Related Questions:

1. How do you solve exponential equations that involve different bases?
2. Can you solve \(2^x + 5 = 13\) using a logarithm?
3. What is the power rule of logarithms, and how is it applied?
4. How does the change of base formula work for logarithms other than base 10 or \(e\)?
5. What are the differences between natural logarithms (\(\ln\)) and common logarithms (\(\log\))?

### Tip:
When solving equations involving exponents, always consider using logarithms to simplify and isolate the variable.

Explain how to solve 3^x - 4 = 6 using the change of base formula log_b(y) = log_k(y) / log_k(b). Include the solution for x in your answer. Round your answer to the nearest thousandth.

Learn how to solve the exponential equation 3^x - 4 = 6 using the change of base formula. This solution includes a step-by-step approach to applying logarithms and the change of base theorem to find the value of x. Suitable for high school students in Grades 10-12.

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