https://file.aiquickdraw.com/mathbot/file/1725519089338yejbh6n4.jpg

The equation shown in the image is:

\[
2 \log_4 4^7 = x
\]

Let's simplify and solve for \( x \).

### Step 1: Simplify the logarithmic expression
We know from the properties of logarithms that:

\[
\log_b (a^n) = n \log_b a
\]

So, apply this rule to \( \log_4 4^7 \):

\[
\log_4 4^7 = 7 \log_4 4
\]

Since \( \log_4 4 = 1 \) (because \( 4^1 = 4 \)), we have:

\[
\log_4 4^7 = 7 \times 1 = 7
\]

### Step 2: Substitute back into the equation
Now the equation becomes:

\[
2 \times 7 = x
\]

\[
x = 14
\]

Thus, the solution is \( x = 14 \).

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Would you like more details or have any questions?

Here are 5 related questions to further your understanding:
1. What are the common properties of logarithms, and how do they help simplify expressions?
2. How does the base of a logarithm affect its value, and why is \( \log_b b = 1 \)?
3. What are the steps for solving equations involving logarithms with different bases?
4. How can exponential and logarithmic functions be used to solve real-world problems?
5. Can you solve \( 3 \log_2 (x^4) = 12 \) using similar properties of logarithms?

**Tip:** Always look for ways to simplify logarithmic expressions using known properties like the power rule or change of base formula.

Learn how to solve the logarithmic equation 2 log_4 4^7 = x by simplifying using properties of logarithms and exponents. This problem is suitable for students in Grades 8-10.

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