Solve : `3^(2x-1)=4^(x+2)`

Options

A. `2.774`
B. `3.774`
C. `1.774`
D. `4.774`

Answer

To solve the equation \( 3^{2x - 1} = 4^{x + 2} \), we can follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the logarithm of both sides of the equation. We can use any logarithm, but for this solution, we will use the natural logarithm (ln) for simplicity.

\[ \ln(3^{2x - 1}) = \ln(4^{x + 2}) \]

### Step 2: Apply the power rule of logarithms
Using the property of logarithms that states \( \ln(a^b) = b \cdot \ln(a) \), we can simplify both sides:

\[ (2x - 1) \cdot \ln(3) = (x + 2) \cdot \ln(4) \]

### Step 3: Expand both sides
Now, we will distribute the logarithms on both sides:

\[ 2x \cdot \ln(3) - \ln(3) = x \cdot \ln(4) + 2 \cdot \ln(4) \]

### Step 4: Rearrange the equation
Next, we want to get all the terms involving \( x \) on one side and the constant terms on the other side. We can rearrange the equation as follows:

\[ 2x \cdot \ln(3) - x \cdot \ln(4) = 2 \cdot \ln(4) + \ln(3) \]

### Step 5: Factor out \( x \)
Now, we can factor out \( x \) from the left side:

\[ x (2 \cdot \ln(3) - \ln(4)) = 2 \cdot \ln(4) + \ln(3) \]

### Step 6: Solve for \( x \)
To isolate \( x \), we divide both sides by \( (2 \cdot \ln(3) - \ln(4)) \):

\[ x = \frac{2 \cdot \ln(4) + \ln(3)}{2 \cdot \ln(3) - \ln(4)} \]

### Step 7: Calculate the value of \( x \)
Now we can calculate the numerical value of \( x \) using a calculator:

Using approximate values:
- \( \ln(3) \approx 1.0986 \)
- \( \ln(4) \approx 1.3863 \)

Substituting these values:

\[ x \approx \frac{2 \cdot 1.3863 + 1.0986}{2 \cdot 1.0986 - 1.3863} \]
\[ x \approx \frac{2.7726 + 1.0986}{2.1972 - 1.3863} \]
\[ x \approx \frac{3.8712}{0.8109} \approx 4.78 \]

### Final Answer:
Thus, the solution to the equation \( 3^{2x - 1} = 4^{x + 2} \) is approximately:

\[ x \approx 4.78 \]

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