# Use the properties of logarithms to expand the expression as

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume the variable is positive.)
$$\displaystyle{\ln{{\left({\frac{{{x}}}{{\sqrt{{{x}^{{2}}+{1}}}}}}\right)}}}$$

• Questions are typically answered in as fast as 30 minutes

### Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

Eprint
$$\displaystyle\text{logarithmic expression}{\ln{{\left({\frac{{{x}}}{{\sqrt{{{x}^{{2}}+{1}}}}}}\right)}}}$$
We know that,$$\displaystyle{\ln{{\left({\frac{{{a}}}{{{b}}}}\right)}}}={\log{{a}}}-{\log{{b}}}$$
$$\displaystyle{\ln{{\left({\frac{{{x}}}{{\sqrt{{{x}^{{2}}+{1}}}}}}\right)}}}$$
$$\displaystyle{\ln{{\left({x}\right)}}}-{\ln{{\left(\sqrt{{{x}^{{2}}+{1}}}\right)}}}$$
$$\displaystyle{\ln{{\left({x}\right)}}}-{{\ln{{\left(\sqrt{{{x}^{{2}}+{1}}}\right)}}}^{{\frac{{{1}}}{{{2}}}}}}$$
$$\displaystyle{\ln{{\left({x}\right)}}}-{\frac{{{1}}}{{{2}}}}{\ln{{\left({x}^{{2}}+{1}\right)}}}$$
$$\displaystyle\because{{\log{{a}}}^{{m}}=}{m}{\log{{a}}}$$
$$\displaystyle{\ln{{\left({x}\right)}}}-{\frac{{{1}}}{{{2}}}}{\ln{{\left({x}^{{2}}+{1}\right)}}}$$
###### Have a similar question?
James Obrien
Help me find a solution please, I just can't solve