Question

# Write the expression as a sum and/or difference of logarithms.

Factors and multiples

Write the expression as a sum and/or difference of logarithms. Express powers as factors.
$$\displaystyle{\ln{{\left({x}^{{{15}}}\sqrt{{{7}-{x}}}\right)}}},{0}{<}{x}{<}{7}$$
$$\displaystyle{\ln{{\left({x}^{{{15}}}\sqrt{{{7}-{x}}}\right)}}}=?$$

2021-08-07

Step 1
Using formula:
1) $$\displaystyle{\log{{\left({A},{B}\right)}}}={\log{{A}}}+{\log{{B}}}$$
2) $$\displaystyle{{\log{{A}}}^{{{n}}}=}{n}{\log{{A}}}$$
root means power $$\displaystyle={\frac{{{1}}}{{{2}}}}$$
$$\displaystyle\sqrt{{{p}}}={p}^{{\frac{{1}}{{2}}}}$$
Step 2
$$\displaystyle{\ln{{\left({x}^{{{15}}}\sqrt{{{7}-{x}}}\right)}}},{0}{<}{x}{<}{7}$$
$$\displaystyle={\ln{{\left({x}^{{{15}}}\right)}}}+{\ln{{\left(\sqrt{{{7}-{x}}}\right)}}}$$
$$\displaystyle={{\ln{{x}}}^{{{15}}}+}{{\ln{{\left({7}-{x}\right)}}}^{{\frac{{1}}{{2}}}}}$$
$$\displaystyle={15}{\ln{{x}}}+{\frac{{{1}}}{{{2}}}}{\ln{{\left({7}-{x}\right)}}}$$
hence, $$\displaystyle{\ln{{\left({x}^{{{15}}}\sqrt{{{7}-{x}}}\right)}}}={15}{\ln{{x}}}+{\frac{{{1}}}{{{2}}}}{\ln{{\left({7}-{x}\right)}}}$$