Evaluate log$_2$ 8 – log$_3$ $\frac{1}{9}$

To solve this expression, we need to apply the fundamental laws of logarithms. Let's break it down into two parts.

1. Analyze the question

We are asked to evaluate: $\log_2 8 - \log_3 \frac{1}{9}$

2. Explain the concepts

• Logarithm Definition: $\log_b a = x$ means $b^x = a$.
• Power Rule: $\log_b (x^n) = n \cdot \log_b x$.
• Negative Exponent Rule: $\frac{1}{x^n} = x^{-n}$.

3. Show the solution

Step 1: Evaluate $\log_2 8$ We know that $8$ can be written as a power of $2$: $8 = 2 \times 2 \times 2 = 2^3$ So, $\log_2 8 = \log_2 (2^3)$ Using the power rule, this becomes: $3 \cdot \log_2 2$ Since $\log_b b = 1$, then $3 \cdot 1 = \mathbf{3}$.

Step 2: Evaluate $\log_3 \frac{1}{9}$ We know that $9 = 3^2$. Therefore, $\frac{1}{9} = \frac{1}{3^2}$. Using the negative exponent rule, $\frac{1}{3^2} = 3^{-2}$. So, $\log_3 \frac{1}{9} = \log_3 (3^{-2})$ Using the power rule, this becomes: $-2 \cdot \log_3 3$ Since $\log_3 3 = 1$, then $-2 \cdot 1 = \mathbf{-2}$.

Step 3: Combine the results Now, substitute these values back into the original expression: $\log_2 8 - \log_3 \frac{1}{9} = 3 - (-2)$ $3 + 2 = 5$

4. Identify the answer

Comparing our result to the given options: A. -1 1/2 B. -1 C. 1 D. 5

The correct option is D.

5. Exam Tip

When you see fractions in logarithms (like $1/9$), immediately think of negative exponents. Whenever you see $\log_b (\frac{1}{x})$, it is almost always going to result in a negative value. Also, always remember that $\log_b b = 1$; it is the most useful shortcut in JAMB mathematics!