If `log_(8)m + log_(6) (1)/(6) = (2)/(3),` then m is equal to

Options

A. 24
B. 18
C. 12
D. 4

Answer

To solve the equation \( \log_{8} m + \log_{6} \left(\frac{1}{6}\right) = \frac{2}{3} \), we will follow these steps:

### Step 1: Simplify the logarithmic expression
We know that \( \log_{6} \left(\frac{1}{6}\right) \) can be simplified. The logarithm of a reciprocal can be expressed as:
\[ \log_{b} \left(\frac{1}{x}\right) = -\log_{b} x \]
Thus, we have:
\[ \log_{6} \left(\frac{1}{6}\right) = -\log_{6} 6 \]
Since \( \log_{6} 6 = 1 \), we get:
\[ \log_{6} \left(\frac{1}{6}\right) = -1 \]

### Step 2: Substitute back into the equation
Now we substitute this back into the original equation:
\[ \log_{8} m - 1 = \frac{2}{3} \]

### Step 3: Isolate \( \log_{8} m \)
To isolate \( \log_{8} m \), we add 1 to both sides:
\[ \log_{8} m = \frac{2}{3} + 1 \]
Converting 1 to a fraction gives us:
\[ \log_{8} m = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} \]

### Step 4: Convert the logarithmic equation to exponential form
Using the definition of logarithms, we can convert this to exponential form:
\[ m = 8^{\frac{5}{3}} \]

### Step 5: Simplify \( 8^{\frac{5}{3}} \)
We can express 8 as \( 2^3 \):
\[ m = (2^3)^{\frac{5}{3}} = 2^{3 \cdot \frac{5}{3}} = 2^5 \]
Calculating \( 2^5 \):
\[ m = 32 \]

Thus, the value of \( m \) is \( 32 \).

### Summary of Steps:
1. Simplify \( \log_{6} \left(\frac{1}{6}\right) \) to get \(-1\).
2. Substitute back into the equation to isolate \( \log_{8} m \).
3. Convert the logarithmic equation to exponential form.
4. Simplify \( 8^{\frac{5}{3}} \) to find \( m \).