In the real numbers, what is the solution of the equation `4^(x+2)=8^(2x-1)`?

Options

A. `-(7)/(4)`
B. `-(1)/(4)`
C. `(5)/(4)`
D. `(7)/(4)`

Answer

To solve the equation \(4^{(x+2)} = 8^{(2x-1)}\), we will follow these steps:

### Step 1: Rewrite the bases
We can express both sides of the equation in terms of base 2. We know that:
- \(4 = 2^2\)
- \(8 = 2^3\)

Thus, we can rewrite the equation as:
\[ (2^2)^{(x+2)} = (2^3)^{(2x-1)} \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify both sides:
\[ 2^{2(x+2)} = 2^{3(2x-1)} \]

### Step 3: Simplify the exponents
Now, we simplify the exponents on both sides:
\[ 2^{2x + 4} = 2^{6x - 3} \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 2x + 4 = 6x - 3 \]

### Step 5: Solve for \(x\)
Now, we will solve for \(x\):
1. Rearranging the equation gives:
\[ 4 + 3 = 6x - 2x \]
\[ 7 = 4x \]
2. Dividing both sides by 4:
\[ x = \frac{7}{4} \]

### Final Answer
Thus, the solution to the equation \(4^{(x+2)} = 8^{(2x-1)}\) is:
\[ \boxed{\frac{7}{4}} \]