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Question Number 160939 by mathlove last updated on 09/Dec/21

$$log\underset{ab}{x}=?iflog\underset{a}{x}=30andlog\underset{b}{x}=70$$

Commented by blackmamba last updated on 09/Dec/21

$$\{\begin{array}{l}a=x^{\frac{1}{30}}\\ b=x^{\frac{1}{70}}\end{array}\Rightarrow \log {}_{ab}\left(x\right)=\log {}_{x^{\frac{1}{30}+\frac{1}{70}}}\left(x\right)=\frac{1}{\frac{1}{30}+\frac{1}{70}}=21$$

Answered by qaz last updated on 09/Dec/21

$${\log }_{\mathrm{ab}}\mathrm{x}=\frac{\mathrm{lnx}}{\ln \left(\mathrm{ab}\right)}=\frac{\mathrm{lnx}}{\mathrm{lna}+\mathrm{lnb}}=\frac{1}{\frac{\mathrm{lna}}{\mathrm{lnx}}+\frac{\mathrm{lnb}}{\mathrm{lnx}}}=\frac{1}{\frac{1}{{\log }_{\mathrm{a}}\mathrm{x}}+\frac{1}{{\log }_{\mathrm{b}}\mathrm{x}}}=\frac{1}{\frac{1}{30}+\frac{1}{70}}=21$$

Answered by cortano last updated on 09/Dec/21

$$\{\begin{array}{l}\log {}_{\mathrm{a}}\left(\mathrm{x}\right)=30\\ \log {}_{\mathrm{b}}\left(\mathrm{x}\right)=70\end{array}$$$$\log {}_{\mathrm{ab}}\left(\mathrm{x}\right)=\frac{1}{\log {}_{\mathrm{x}}\left(\mathrm{ab}\right)}=\frac{1}{\log {}_{\mathrm{x}}\left(\mathrm{a}\right)+\log {}_{\mathrm{x}}\left(\mathrm{b}\right)}$$$$=\frac{1}{\frac{1}{30}+\frac{1}{70}}=\frac{30\times 70}{100}=21$$

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