What is the real value of x in the eqaution, `log_2 80- log_2 5 = log_3 x`?

Options

A. 8
B. 48
C. 75
D. 81

Answer

To solve the equation \( \log_2 80 - \log_2 5 = \log_3 x \), we will follow these steps:

### Step 1: Use the property of logarithms
We know that \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \). Therefore, we can rewrite the left-hand side of the equation:

\[ \log_2 80 - \log_2 5 = \log_2 \left( \frac{80}{5} \right) \]

### Step 2: Simplify the fraction
Now, we simplify \( \frac{80}{5} \):

\[ \frac{80}{5} = 16 \]

So, we can rewrite the equation as:

\[ \log_2 16 = \log_3 x \]

### Step 3: Express 16 as a power of 2
We know that \( 16 = 2^4 \). Thus, we can write:

\[ \log_2 16 = \log_2 (2^4) \]

### Step 4: Use the property of logarithms
Using the property \( \log_a (b^n) = n \log_a b \), we have:

\[ \log_2 (2^4) = 4 \log_2 2 \]

Since \( \log_2 2 = 1 \), this simplifies to:

\[ \log_2 16 = 4 \]

### Step 5: Set the equations equal
Now we have:

\[ 4 = \log_3 x \]

### Step 6: Convert the logarithmic equation to exponential form
Using the property that if \( \log_a b = c \), then \( a^c = b \), we can rewrite the equation as:

\[ 3^4 = x \]

### Step 7: Calculate \( 3^4 \)
Calculating \( 3^4 \):

\[ 3^4 = 81 \]

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

\[ \boxed{81} \]

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