Without using a calculator, simplify log4 (1/16).

Without using a calculator, simplify log4 (1/16).

The correct Answer and Explanation is:

Let’s break this down step by step.

Simplification:

The given logarithmic expression is:

log⁡4(116)\log_4 \left(\frac{1}{16}\right)

1. Express 16 in terms of base 4:

16=4216 = 4^2

So we rewrite the expression as:

log⁡4(142)\log_4 \left(\frac{1}{4^2}\right)

2. Apply the logarithmic rule for reciprocals:

log⁡b(1x)=−log⁡b(x)\log_b \left(\frac{1}{x}\right) = -\log_b (x)

Using this rule, we get:

−log⁡4(42)-\log_4 \left(4^2\right)

3. Apply the power rule:

log⁡b(xn)=nlog⁡b(x)\log_b (x^n) = n \log_b (x)

Thus:

−2log⁡4(4)-2 \log_4 (4)

4. Since log⁡4(4)=1\log_4 (4) = 1, the expression simplifies further to:

−2×1=−2-2 \times 1 = -2

Explanation:

This solution follows fundamental logarithm properties. First, we converted 16 into its equivalent exponent form based on the given logarithm base (4). Recognizing that the logarithm of a fraction is the negative of the logarithm of its denominator, we applied the reciprocal rule. Then, using the logarithm power rule, we extracted the exponent from within the logarithmic expression. Finally, we computed log⁡4(4)\log_4 (4), which equals 1 since the logarithm of a number with its own base is always 1.

Understanding logarithms is crucial in mathematical applications, including solving exponential equations and analyzing growth trends in real-world scenarios. Logarithmic properties simplify complex expressions and enable efficient computations without requiring a calculator.

So the final answer is -2! Let me know if you’d like further clarification.