Question

what is the value of \\(\log_{0.5} 16\\)?
\\(\circ\\) \\(-4.00\\)
\\(\circ\\) \\(-0.25\\)
\\(\circ\\) \\(1.51\\)
\\(\circ\\) \\(2.41\\)

Explanation:

Step1: Recall logarithm change of base formula

The change of base formula for logarithms is $\log_{a}b=\frac{\log_{c}b}{\log_{c}a}$ (where $c>0,c
eq1$). We can also use the property of logarithms related to exponents. Let's first express $0.5$ as a power of $2$, $0.5 = 2^{-1}$, and $16=2^{4}$.

Step2: Use logarithm property $\log_{a^m}b^n=\frac{n}{m}\log_{a}b$

We know that $\log_{0.5}16=\log_{2^{-1}}2^{4}$. By the property $\log_{a^m}b^n = \frac{n}{m}\log_{a}b$, here $a = 2$, $m=- 1$, $b = 2$, $n = 4$. So $\log_{2^{-1}}2^{4}=\frac{4}{-1}\log_{2}2$. Since $\log_{2}2 = 1$, then $\frac{4}{-1}\times1=-4$.
Alternatively, using change of base formula with base 10: $\log_{0.5}16=\frac{\log_{10}16}{\log_{10}0.5}$. We know that $\log_{10}16\approx1.2041$, $\log_{10}0.5\approx - 0.3010$. Then $\frac{1.2041}{-0.3010}\approx - 4$.

Answer:

-4.00