Solve for x:
` log_(4) (2x+3) =(3)/(2)`

Answer

To solve the equation \( \log_{4}(2x + 3) = \frac{3}{2} \), we will follow these steps:

### Step 1: Convert the logarithmic equation to its exponential form.
The logarithmic equation \( \log_{4}(2x + 3) = \frac{3}{2} \) can be rewritten in exponential form as:
\[ 2x + 3 = 4^{\frac{3}{2}} \]

### Step 2: Calculate \( 4^{\frac{3}{2}} \).
We know that \( 4^{\frac{3}{2}} \) can be calculated as follows:
\[ 4^{\frac{3}{2}} = (4^{\frac{1}{2}})^3 = (2)^3 = 8 \]
So, we have:
\[ 2x + 3 = 8 \]

### Step 3: Isolate \( x \).
Now, we will isolate \( x \) by subtracting 3 from both sides:
\[ 2x = 8 - 3 \]
\[ 2x = 5 \]

### Step 4: Solve for \( x \).
Next, we divide both sides by 2:
\[ x = \frac{5}{2} \]

### Final Answer:
Thus, the solution for \( x \) is:
\[ x = 2.5 \]

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