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Question Number 17748 by ibraheem160 last updated on 10/Jul/17

$$f{p}^2=qr,provethathat{log}_r^p+{log}_{q=}^p$$$$2{log}_q^p{log}_r^p$$

Commented by tawa tawa last updated on 10/Jul/17

$$\frac{\mathrm{logp}}{\mathrm{logr}}+\frac{\mathrm{logp}}{\mathrm{logq}}$$$$=\frac{\mathrm{logplogq}+\mathrm{logplogr}}{\mathrm{logrlogq}}$$$$=\frac{\mathrm{logp}[\mathrm{logq}+\mathrm{logr}]}{\mathrm{logrlogq}}$$$$=\frac{\mathrm{logp}\left(\mathrm{logqr}\right)}{\mathrm{logrlogq}}$$$$\mathrm{from}{\mathrm{p}}^2=\mathrm{qr}$$$$=\frac{{\mathrm{logplogp}}^2}{\mathrm{logrlogq}}$$$$=\frac{\mathrm{logp}}{\mathrm{logr}}\times \frac{{\mathrm{logp}}^2}{\mathrm{logq}}$$$$={\log }_{\mathrm{r}}\mathrm{p}\times {2\log }_{\mathrm{q}}\mathrm{p}$$$$={2\log }_{\mathrm{q}}{\mathrm{plog}}_{\mathrm{r}}\mathrm{p}$$$$\mathrm{QED}$$

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