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Question Number 191392 by Mingma last updated on 23/Apr/23

Answered by Frix last updated on 23/Apr/23

$$\frac{\sqrt{\ln n}}{\sqrt{\ln b}}=\frac{\ln n}{2\ln b}\Rightarrow \ln n=4\ln b$$$$\frac{b\ln n}{\ln b}=\frac{\ln bn}{\ln b}\Rightarrow \ln n=\frac{\ln b}{b-1}$$$$\Rightarrow$$$$4\ln b=\frac{\ln b}{b-1}\Rightarrow b=\frac{5}{4}\Rightarrow n=\frac{625}{256}$$

Commented by Mingma last updated on 24/Apr/23

Excellent

Answered by cortano12 last updated on 24/Apr/23

$$\mathrm{let}\log {}_{\mathrm{b}}\mathrm{n}=\mathrm{x}$$$$\{\begin{array}{l}\sqrt{\mathrm{x}}=\frac{1}{2}\mathrm{x}\Rightarrow 4\mathrm{x}={\mathrm{x}}^2\Rightarrow \{\begin{array}{l}\mathrm{x}=0\\ \mathrm{x}=4\end{array}\\ \mathrm{bx}=1+\mathrm{x}\Rightarrow 4\mathrm{b}=5;\mathrm{b}=\frac{5}{4}\end{array}$$$$\iff \log {}_{\frac{5}{4}}\mathrm{n}=4;\mathrm{n}={\left(\frac{5}{4}\right)}^4=\frac{625}{256}$$

Commented by Mingma last updated on 24/Apr/23

Excellent

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