https://file.aiquickdraw.com/mathbot/file/1726018523915wrdcr1wt.jpg

The equation in the image is:

\[
6^{x - 2} = 64^6
\]

We need to solve for \( x \).

### Step 1: Express both sides with the same base
We observe that \( 64 \) can be rewritten as a power of \( 4 \), since:

\[
64 = 4^3
\]

So, we can rewrite the equation as:

\[
6^{x - 2} = (4^3)^6
\]

Simplify the right-hand side:

\[
(4^3)^6 = 4^{18}
\]

Now the equation becomes:

\[
6^{x - 2} = 4^{18}
\]

### Step 2: Solve using logarithms
Since the bases are different (6 and 4), we can take the natural logarithm (or logarithm to any base) of both sides to isolate \( x \):

\[
\ln(6^{x - 2}) = \ln(4^{18})
\]

Using logarithmic properties, this becomes:

\[
(x - 2) \ln(6) = 18 \ln(4)
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \):

\[
x - 2 = \frac{18 \ln(4)}{\ln(6)}
\]

\[
x = 2 + \frac{18 \ln(4)}{\ln(6)}
\]

We can calculate this value numerically:

Let me compute this for you.The solution to the equation \(6^{x - 2} = 64^6\) is approximately:

\[
x \approx 15.93
\]

Would you like further details or explanations on any part of this process? Here are some related questions:

1. How do logarithmic properties help in solving exponential equations?
2. What happens if both sides of the equation cannot be expressed with the same base?
3. Could you solve this equation using a different method, such as trial and error?
4. How can you verify if the solution is correct?
5. How do natural logarithms differ from logarithms of other bases?

**Tip:** When solving exponential equations, it's often helpful to rewrite both sides with the same base if possible, as it simplifies the equation.

Solve the equation 6^(x - 2) = 64^6 for x.

Learn how to solve the exponential equation 6^(x - 2) = 64^6 step by step using logarithms. This problem involves key concepts like rewriting expressions and using logarithmic properties to find the solution. Suitable for high school students in Grades 10-12.

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