If ` 3^(x) = 4^(x-1)` , then x =

Options

A. ` (2 log_(3) 2)/(2log_(3) 2-1)`
B. ` 2/(2-log_(2)3)`
C. ` 1/(1-log_(4)3)`
D. ` (2 log_(2)3)/(2 log_(2) 3-1)`

Explanation

To solve the equation \( 3^x = 4^{x-1} \), 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.

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

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

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

### Step 3: Expand the right side
Now, we expand the right side of the equation:

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

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

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

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

\[ x (\ln(3) - \ln(4)) = -\ln(4) \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):

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

### Step 7: Simplify the expression
We can simplify this further using the properties of logarithms:

\[ x = \frac{\ln(4)}{\ln(4) - \ln(3)} = \frac{\ln(4)}{\ln\left(\frac{4}{3}\right)} \]

### Final Answer
Thus, the value of \( x \) is:

\[ x = \frac{\ln(4)}{\ln\left(\frac{4}{3}\right)} \]