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Question Number 145119 by imjagoll last updated on 02/Jul/21

$$\frac{5^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(3\right)}}=?$$

Answered by Rasheed.Sindhi last updated on 03/Jul/21

$$\frac{5^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(3\right)}}=?$$$$\log {}_{\frac{5}{3}}\left(5\right)=a;\log {}_{\frac{5}{3}}\left(3\right)=b$$$${\left(\frac{5}{3}\right)}^a=5;{\left(\frac{5}{3}\right)}^b=3$$$$\frac{{\left(\frac{5}{3}\right)}^a}{{\left(\frac{5}{3}\right)}^b}=\frac{5}{3}$$$${\left(\frac{5}{3}\right)}^{a-b}={\left(\frac{5}{3}\right)}^1$$$$a-b=1\to b=a-1$$$$\frac{5^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(3\right)}}=\frac{5^a}{3^b}=\frac{5^a}{3^{a-1}}$$$$=3{\left(\frac{5}{3}\right)}^a$$$$=3{\left(\frac{5}{3}\right)}^{\log {}_{\frac{5}{3}}\left(5\right)}=3\left(5\right)=15$$$$$$$$\overline{)\begin{array}{c}{\mathrm{b}}^{{\log }_{\mathrm{b}}\left(\mathrm{a}\right)}=\mathrm{a}\end{array}}$$

Answered by liberty last updated on 03/Jul/21

$$\mathrm{let}\{\begin{array}{l}\log {}_{\frac{5}{3}}\left(5\right)=\mathrm{x}\Rightarrow 5^{x-1}=3^x\\ \log {}_{\frac{5}{3}}\left(3\right)=y\Rightarrow 5^y=3^{y+1}\end{array}$$$$\{\begin{array}{l}5=3^{\frac{x}{x-1}}\\ {\left(3^{\frac{x}{x-1}}\right)}^y=3^{y+1}\end{array}\Rightarrow \frac{xy}{x-1}=y+1$$$$\Rightarrow xy=xy+x-y-1$$$$\Rightarrow x=y+1.sothevalueof$$$$\frac{5^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(3\right)}}=\frac{5^x}{3^y}=\frac{5.5^y}{3^y}$$$$\Rightarrow 5\times {\left(\frac{5}{3}\right)}^y=5\times 3=15$$

Answered by Rasheed.Sindhi last updated on 03/Jul/21

$$\frac{5^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(3\right)}}$$$$$$$$=\frac{\frac{5^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(5\right)}}\times 3^{\log {}_{\frac{5}{3}}\left(5\right)}}{3^{\log {}_{\frac{5}{3}}\left(3\right)}}$$$$$$$$={\left(\frac{5}{3}\right)}^{\log {}_{\frac{5}{3}}\left(5\right)}\times 3^{\log {}_{\frac{5}{3}}\left(5\right)-\log {}_{\frac{5}{3}}\left(3\right)}$$$$\overline{)\begin{array}{c}{\mathrm{b}}^{{\log }_{\mathrm{b}}\mathrm{a}}=\mathrm{a}\end{array}}$$$$=5\times 3^{{\log }_{\frac{5}{3}}\left(\frac{5}{3}\right)}=5\times 3^1=15$$$$\overline{)\begin{array}{c}{\log }_{\mathrm{a}}\mathrm{a}=1\end{array}}$$

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