If `7^(x-1) = 6x` , find x .

Options

A. `-13.2`
B. `0.08`
C. `0.22`
D. 12.6

Explanation

To solve the equation \( 7^{(x-1)} = 6x \), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ 7^{(x-1)} = 6x \]

### Step 2: Take the Natural Logarithm of Both Sides
Next, we take the natural logarithm (ln) of both sides:
\[ \ln(7^{(x-1)}) = \ln(6x) \]

### Step 3: Apply the Power Rule of Logarithms
Using the property of logarithms that states \( \ln(a^b) = b \cdot \ln(a) \), we can simplify the left side:
\[ (x-1) \cdot \ln(7) = \ln(6x) \]

### Step 4: Expand the Left Side
Now, we expand the left-hand side:
\[ x \cdot \ln(7) - \ln(7) = \ln(6x) \]

### Step 5: Rearrange the Equation
Next, we rearrange the equation to isolate terms involving \( x \):
\[ x \cdot \ln(7) - \ln(6x) = \ln(7) \]

### Step 6: Rewrite \( \ln(6x) \)
We can express \( \ln(6x) \) as:
\[ \ln(6) + \ln(x) \]
So the equation becomes:
\[ x \cdot \ln(7) - (\ln(6) + \ln(x)) = \ln(7) \]

### Step 7: Rearrange Again
Rearranging gives us:
\[ x \cdot \ln(7) - \ln(6) - \ln(x) = \ln(7) \]
\[ x \cdot \ln(7) - \ln(x) = \ln(7) + \ln(6) \]

### Step 8: Isolate \( x \)
We can now isolate \( x \):
\[ x \cdot \ln(7) - \ln(6) = \ln(7) + \ln(6) \]
\[ x \cdot \ln(7) = \ln(7) + \ln(6) + \ln(6) \]
\[ x = \frac{\ln(7) + \ln(6)}{\ln(7) - \ln(6)} \]

### Step 9: Calculate the Values
Using a calculator, we find:
- \( \ln(7) \approx 1.94591 \)
- \( \ln(6) \approx 1.79176 \)

Substituting these values:
\[ x = \frac{1.94591}{1.94591 - 1.79176} = \frac{1.94591}{0.15415} \approx 12.63 \]

### Final Answer
Thus, the value of \( x \) is approximately:
\[ x \approx 12.63 \]