Question

Determine whether the equation is true or false. Where​ possible, show work to support your conclusion. If the statement is​ false, make the necessary​ change(s) to produce a true statement.
x log Subscript 4 Baseline 4 Superscript x Baseline equals x squared
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The equation is false. The correct equation is x log Subscript 4 Baseline 4 Superscript x ​=
  
enter your response here.
B.
The equation is true.

Answer 1

To determine whether the equation \( x \log_4 (4^x) = x^2 \) is true or false, we can start by simplifying the left side of the equation.

1. Using the logarithm property, we know that \( \log_b (a^c) = c \cdot \log_b (a) \). Therefore:
\[
\log_4 (4^x) = x \cdot \log_4 (4)
\]

2. Since \( \log_4 (4) = 1 \) (because \( 4^1 = 4 \)), we have:
\[
\log_4 (4^x) = x \cdot 1 = x
\]

3. Substituting this back into the left side of the original equation gives us:
\[
x \log_4 (4^x) = x \cdot x = x^2
\]

Thus, the left-hand side simplifies to \( x^2 \).

Now we can rewrite the original equation based on our simplification:
\[
x^2 = x^2
\]

Since this equation holds true for all \( x \), we can conclude:

### Conclusion
The statement \( x \log_4 (4^x) = x^2 \) is **true**.

So the correct choice is:
B. The equation is true.