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Question Number 8720 by www.matikzone.tk last updated on 24/Oct/16

Commented by prakash jain last updated on 24/Oct/16

$$whatismeantby{a}_{\log c}$$

Commented by www.matikzone.tk last updated on 24/Oct/16

Commented by www.matikzone.tk last updated on 24/Oct/16

$$\mathrm{the}\mathrm{solution}a=3b=4c=8\mathrm{and}d=9$$$$\mathrm{how}\mathrm{to}\mathrm{get}\mathrm{it}?$$

Commented by prakash jain last updated on 24/Oct/16

$$d=9,a=3$$$$d-a=6$$$$\mathrm{Does}\mathrm{not}\mathrm{satisfy}d-a=7$$$$\mathrm{Should}\mathrm{be}a=2,b=4,c=8,d=9$$

Commented by prakash jain last updated on 24/Oct/16

$$\frac{\ln 81}{\ln \left[\frac{\ln c}{\ln \left[\frac{\ln b}{\ln \left[\frac{\ln d}{\ln \frac{\ln c}{\ln a}}\right]}\right]}\right]}=b$$$$ab=c$$$$d=a+7$$$$\mathrm{This}\mathrm{gives}\mathrm{us}4\mathrm{equations}\mathrm{with}$$$$\mathrm{three}\mathrm{unknowns}.$$$$a=2,b=4,c=8,d=9$$$$\frac{\ln 81}{\ln \left[\frac{\ln 8}{\ln \left[\frac{\ln 4}{\ln \left[\frac{\ln 9}{\ln \frac{\ln 8}{\ln 2}}\right]}\right]}\right]}=$$$$\frac{\ln 81}{\ln \left[\frac{\ln 8}{\ln \left[\frac{\ln 4}{\ln \left[\frac{\ln 9}{\ln 3}\right]}\right]}\right]}=$$$$\frac{\ln 81}{\ln \left[\frac{\ln 8}{\ln \left[\frac{\ln 4}{\ln 2}\right]}\right]}=\frac{\ln 81}{\ln \left[\frac{\ln 8}{\ln 2}\right]}=\frac{\ln 81}{\ln 3}=4=b$$$$\mathrm{So}\mathrm{this}\mathrm{satisfies}\mathrm{the}\mathrm{equation}.$$$$\mathrm{However}\mathrm{given}\mathrm{the}\mathrm{fact}\mathrm{there}\mathrm{are}$$$$\mathrm{only}3\mathrm{equation}\mathrm{with}4\mathrm{unknown}$$$$\mathrm{we}\mathrm{can}\mathrm{assume}\mathrm{value}\mathrm{for}\mathrm{one}$$$$\mathrm{of}\mathrm{the}\mathrm{variable}\mathrm{and}\mathrm{calculate}\mathrm{theothers}.$$$$\mathrm{So}\mathrm{i}\mathrm{start}\mathrm{by}\mathrm{assuming}b=e$$$$\mathrm{then}\mathrm{i}\mathrm{calculated}\mathrm{the}\mathrm{following}$$$$\mathrm{using}\mathrm{wolfram}\mathrm{alpha}\mathrm{caculator}.$$$$b=e=2.7182818284590..$$$$a=1.7296241323821...$$$$c=4.70160606890820..$$$$d=8.7296241323821...$$$$abcd=196.96921294...$$$$\mathrm{So}\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{unique}\mathrm{solution}.$$$$\mathrm{The}\mathrm{above}\mathrm{solution}\mathrm{also}\mathrm{meets}$$$$\mathrm{the}\mathrm{condition}.$$

Answered by prakash jain last updated on 24/Oct/16

$$\mathrm{No}\mathrm{unique}\mathrm{solution}.\mathrm{See}\mathrm{explaination}$$$$\mathrm{in}\mathrm{comments}.$$

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