Question

Rewrite the logarithmic as a ratio of a:
(a) a common logarithm
(b) a natural logarithm

Log X
5

Answer 1

\(\Large log_5x=a\quad ?\)

Answer 2

yea

Answer 3

use the change of base rule, recall -> \(\bf \textit{change of base rule}= log_{\color{red}{ a}}{\color{blue}{ b}}\implies \cfrac{log_{\color{green}{ c}}{\color{blue}{ b}}}{log_{\color{green}{ c}}{\color{red}{ a}}}
\\ \quad \\
and\ {\color{green}{ c}}\textit{ could be any value, so long is the same above and below}\)

Answer 4

so c could be 10?

Answer 5

it could be, "anything" really \(\bf \textit{change of base rule}= log_{\color{red}{ a}}{\color{blue}{ b}}\implies \cfrac{log_{\color{green}{ c}}{\color{blue}{ b}}}{log_{\color{green}{ c}}{\color{red}{ a}}}
\\ \quad \\
\cfrac{log_{\color{green}{ cheese}}{\color{blue}{ b}}}{log_{\color{green}{ cheese}}{\color{red}{ a}}}\qquad \cfrac{log_{\color{green}{ meow}}{\color{blue}{ b}}}{log_{\color{green}{ meow}}{\color{red}{ a}}}\)

10, e, 27, 1,000,000, anything

Answer 6

ok, what about the common or natural log?

Answer 7

sure, recall that \(\bf ln =log_{\color{green}{ e}}\)

Answer 8

\(\bf \textit{change of base rule}= log_{\color{red}{ a}}{\color{blue}{ b}}\implies \cfrac{log_{\color{green}{ c}}{\color{blue}{ b}}}{log_{\color{green}{ c}}{\color{red}{ a}}}
\\ \quad \\
\cfrac{log_{\color{green}{ 10}}{\color{blue}{ b}}}{log_{\color{green}{ 10}}{\color{red}{ a}}}\qquad \cfrac{log_{\color{green}{ e}}{\color{blue}{ b}}}{log_{\color{green}{ e}}{\color{red}{ a}}}\)

Answer 9

oh ok

Answer 10

so just use the change of base rule :)