Question

question
write the expression below as a single logarithm in simplest form.
$4\log_{b} 2 - \log_{b} 8$

Explanation:

Step1: Apply power rule of logarithms

The power rule states that \( n\log_{b}a=\log_{b}a^{n} \). So for \( 4\log_{b}2 \), we have \( 4\log_{b}2 = \log_{b}2^{4}\). Calculating \( 2^{4}=16 \), so \( 4\log_{b}2=\log_{b}16 \).

Step2: Apply quotient rule of logarithms

The quotient rule states that \( \log_{b}a-\log_{b}c = \log_{b}\frac{a}{c} \). Now we have \( \log_{b}16-\log_{b}8 \), so applying the quotient rule, we get \( \log_{b}\frac{16}{8} \).

Step3: Simplify the fraction

Simplify \( \frac{16}{8}=2 \). So the expression becomes \( \log_{b}2 \).

Answer:

\(\log_{b}2\)