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

Options

A. `(2 log_(3) 2)/(2 log_(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)`

Answer

To solve the equation \(3^x = 4^{x-1}\), we will 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) or common logarithm (log).

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

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

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

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

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

### Step 4: Rearrange the equation
Next, we will rearrange the equation to isolate terms involving \(x\):

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

Factoring out \(x\) from the left side gives:

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

### Step 5: Solve for \(x\)
Now, we can solve for \(x\) by dividing both sides by \((\log(3) - \log(4))\):

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

### Step 6: Simplify the expression
We can simplify the expression further. Recall that \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\):

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

### Step 7: Change the base of the logarithm
Using the change of base formula, we can express this in terms of base 3:

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

### Final Expression
This can be expressed in terms of base 2:

\[ x = \frac{2 \cdot \log(2)}{\log(3) - 2 \cdot \log(2)} \]

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

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