Question

Logarithm question

Answer 1

\(\large \color{black}{\begin{align} &If \hspace{.33em}\\~\\
&\dfrac13 \log_{3}M+3\log_{3}N=1+\log_{0.008} 5 \hspace{.33em}\\~\\

&then \hspace{.33em}\\~\\
&(1)\ \ M^{9}=\dfrac{9}{N} \hspace{.33em}\\~\\
&(2)\ \ N^{9}=\dfrac{9}{M} \hspace{.33em}\\~\\
&(3)\ \ M^{3}=\dfrac{3}{N} \hspace{.33em}\\~\\
&(4)\ \ N^{9}=\dfrac{3}{M} \hspace{.33em}\\~\\
\end{align}}\)

Answer 2

\(\large \color{black} \\~\\
\dfrac13 \log_{3}M+3\log_{3}N=1+\log_{0.008} 5 \\~\\
\log_{3}M^{1/3}+\log_{3}N^3=1+\log_{0.008} 5 \\~\\
\log_{3}\left( M^{1/3}\cdot N^3\right)=1+\log_{0.008} 5 \\~\\
\log_{3}\left( M^{1/3}\cdot N^3\right)=1+\log_{1/125} 5 \\~\\
\log_{3}\left( M^{1/3}\cdot N^3\right)=1+(-1/3) \\~\\
\log_{3}\left( M^{1/3}\cdot N^3\right)=2/3 \\~\\
\left( M^{1/3}\cdot N^3\right)=3^{2/3} \\~\\
\left( M^{1/3}\cdot N^3\right)^3=\left(3^{2/3} \right)^3\\~\\
M\cdot N^9 = 3^2

\)

Answer 3

well there's another way to solve this question @mathmath333

Answer 4

which one

Answer 5

the left side is easy to solve that yu would have done by the perls way.

Answer 6

for the write side log base(0.008) (5) you can write it as log base((0.2)^3) (5) which will equal to 1/3 * log base(0.2) 5 now 0.2 can be written as 1/5 that means 1/3 * [log base(1/5) 5] which in turns equals to 1/3 * [log 5/ (log(1/5))] which finally becomes 1/3 * [log 5/ (0-log5)] which gives -1/3

Answer 7

@mathmath333

Answer 8

right*