Harlen Pritchard

2021-08-18

Please, write the logarithm as a ratio of common logarithms and natural logarithms.

${\mathrm{log}}_{3}\left(x\right)$

a) common logarithms

b) natural logarithms

a) common logarithms

b) natural logarithms

Khribechy

Skilled2021-08-19Added 100 answers

There is ${\mathrm{log}}_{3}\left(x\right)={\mathrm{log}}_{10}\left(x\right)\cdot {\mathrm{log}}_{3}\left(10\right)=$

$=\frac{{\mathrm{log}}_{10}\left(x\right)}{{\mathrm{log}}_{10}\left(3\right)}=$

$=\frac{\mathrm{log}\left(x\right)}{\mathrm{log}\left(3\right)}$

Similarly,${\mathrm{log}}_{3}\left(x\right)={\mathrm{log}}_{e}\left(x\right)\cdot {\mathrm{log}}_{3}\left(e\right)=$

$=\frac{{\mathrm{log}}_{e}\left(x\right)}{{\mathrm{log}}_{e}\left(3\right)}=$

$=\frac{\mathrm{ln}\left(x\right)}{\mathrm{ln}\left(3\right)}$

a) common logarithm:$\frac{{\mathrm{log}}_{10}\left(x\right)}{{\mathrm{log}}_{10}\left(3\right)}$

b) natural logarithm:$\frac{{\mathrm{log}}_{e}\left(x\right)}{{\mathrm{log}}_{e}\left(3\right)}$

Similarly,

a) common logarithm:

b) natural logarithm: