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

Options

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

Answer

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

### Step 1: Take the logarithm of both sides
We start with the equation:
\[ 3^x = 4^{x-1} \]
Taking the logarithm (base 10 or natural logarithm) of both sides gives us:
\[ \log(3^x) = \log(4^{x-1}) \]

### Step 2: Apply the logarithmic power rule
Using the property of logarithms that states \(\log(a^b) = b \log(a)\), we can rewrite the equation:
\[ x \log(3) = (x - 1) \log(4) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side gives us:
\[ x \log(3) = x \log(4) - \log(4) \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate terms involving \(x\):
\[ x \log(3) - x \log(4) = -\log(4) \]
Factoring out \(x\) from the left-hand side:
\[ x (\log(3) - \log(4)) = -\log(4) \]

### 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
Using the property of logarithms that states \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\), we can rewrite the denominator:
\[ x = \frac{-\log(4)}{\log\left(\frac{3}{4}\right)} \]

### Step 7: Change the sign
Since \(\log\left(\frac{3}{4}\right)\) is negative (because \(3 < 4\)), we can change the sign:
\[ x = \frac{\log(4)}{\log\left(\frac{4}{3}\right)} \]

### Final Result
Thus, the final value of \(x\) is:
\[ x = \frac{\log(4)}{\log\left(\frac{4}{3}\right)} \]