If `log_(7)x = 4/3`, then the value of x is:

Options

A. 9
B. 27
C. 81
D. none of these

Answer

To solve the equation \( \log_{7}x = \frac{4}{3} \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation in exponential form
The logarithmic equation \( \log_{a}b = c \) can be rewritten in exponential form as \( b = a^c \). Here, \( a = 7 \), \( b = x \), and \( c = \frac{4}{3} \).

So, we rewrite it as:
\[ x = 7^{\frac{4}{3}} \]

### Step 2: Simplify \( 7^{\frac{4}{3}} \)
To simplify \( 7^{\frac{4}{3}} \), we can express it as:
\[ x = \left(7^4\right)^{\frac{1}{3}} \]

### Step 3: Calculate \( 7^4 \)
Now, we need to calculate \( 7^4 \):
\[ 7^4 = 7 \times 7 \times 7 \times 7 = 49 \times 49 = 2401 \]

### Step 4: Take the cube root
Now, we need to find the cube root of \( 2401 \):
\[ x = \sqrt[3]{2401} \]

### Final Result
Thus, the value of \( x \) is:
\[ x = 7^{\frac{4}{3}} \text{ or } x = \sqrt[3]{2401} \]