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We are given the function \( f(x) = 4^x + 3(2^{x+2}) \), and we need to find the value of \( m \) such that \( f(m) = 64 \).

Let's solve this step by step:

1. Start by substituting the given function for \( f(m) = 64 \):

\[
f(m) = 4^m + 3(2^{m+2}) = 64
\]

2. Simplify the expression:

We know that \( 2^2 = 4 \), so we can express \( 4^m \) as \( (2^2)^m = 2^{2m} \). Now the equation becomes:

\[
2^{2m} + 3(2^{m+2}) = 64
\]

3. Simplify the term \( 3(2^{m+2}) \):

\[
3(2^{m+2}) = 3(2^m \cdot 2^2) = 3(2^m \cdot 4) = 12(2^m)
\]

Now the equation is:

\[
2^{2m} + 12(2^m) = 64
\]

4. Let \( y = 2^m \). This substitution simplifies the equation to:

\[
y^2 + 12y = 64
\]

5. Rearrange into a standard quadratic form:

\[
y^2 + 12y - 64 = 0
\]

6. Solve this quadratic equation using the quadratic formula:

\[
y = \frac{-12 \pm \sqrt{12^2 - 4(1)(-64)}}{2(1)}
\]
\[
y = \frac{-12 \pm \sqrt{144 + 256}}{2}
\]
\[
y = \frac{-12 \pm \sqrt{400}}{2}
\]
\[
y = \frac{-12 \pm 20}{2}
\]

This gives us two possible solutions for \( y \):

\[
y = \frac{-12 + 20}{2} = 4 \quad \text{or} \quad y = \frac{-12 - 20}{2} = -16
\]

Since \( y = 2^m \), and \( 2^m \) must be positive, we take \( y = 4 \).

7. Now, solve for \( m \):

\[
2^m = 4 \quad \Rightarrow \quad m = 2
\]

Thus, the value of \( m \) is \( \boxed{2} \).

Would you like more details on any part of this solution?

Here are five related questions to expand your understanding:
1. How would the solution change if the equation was \( f(m) = 128 \) instead of \( 64 \)?
2. What happens if we solve the quadratic equation using factoring instead of the quadratic formula?
3. How can we generalize the function \( f(x) = 4^x + 3(2^{x+2}) \) for any other values of \( f(m) \)?
4. What are the properties of exponents used in simplifying \( 4^m \) to \( 2^{2m} \)?
5. What would the solution look like if \( 2^m = 8 \) instead of \( 4 \)?

**Tip**: When solving exponential equations, always look for ways to express different bases in terms of powers of 2 or 10 for easier manipulation.

Diketahui f(x) = 4^x + 3(2^{x+2}). Jika f(m) = 64, maka nilai m adalah… A. 1 B. 2 C. 3 D. 4 E. 5

Learn how to solve the exponential equation f(x) = 4^x + 3(2^{x+2}) for f(m) = 64 with a step-by-step solution. This problem covers algebraic concepts such as exponential functions and solving quadratic equations. Suitable for high school students.

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