Question

1. multiple choice \\(\log_{9} 64 =\\)\

(a) \\(5 \log_{3} 2\\)\
(b) \\((\log_{3} 8)^{2}\\)\
(c) \\((\ln 64)/(\ln 9)\\)\
(d) \\(2 \log_{9} 32\\)\
(e) \\((\log 64)/9\\)

Explanation:

Step1: Use the change - of - base formula

The change - of - base formula for logarithms is \(\log_{a}b=\frac{\ln b}{\ln a}\).
For \(\log_{9}64\), where \(a = 9\) and \(b = 64\), we have \(\log_{9}64=\frac{\ln64}{\ln9}\).

Step2: Check other options

• Option A:

\(5\log_{3}2=\log_{3}2^{5}=\log_{3}32
eq\log_{9}64\) (since \(\log_{9}64=\frac{\log_{3}64}{\log_{3}9}=\frac{\log_{3}2^{6}}{2}=\frac{6\log_{3}2}{2} = 3\log_{3}2\)).

• Option B:

\((\log_{3}8)^{2}=(\log_{3}2^{3})^{2}=(3\log_{3}2)^{2}=9(\log_{3}2)^{2}
eq\log_{9}64\).

• Option D:

\(2\log_{9}32=\log_{9}32^{2}=\log_{9}1024
eq\log_{9}64\).

• Option E:

\(\frac{\log64}{9}
eq\log_{9}64\) (by the definition of logarithms \(\log_{a}b
eq\frac{\log b}{a}\)).

Answer:

C. \((\ln64)/(\ln9)\)