Question

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

Factors and multiples
ANSWERED
asked 2021-08-06

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

Answers (1)

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, \(\displaystyle{\ln{{\left({x}^{{{15}}}\sqrt{{{7}-{x}}}\right)}}}={15}{\ln{{x}}}+{\frac{{{1}}}{{{2}}}}{\ln{{\left({7}-{x}\right)}}}\)

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