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Question Number 145067 by imjagoll last updated on 02/Jul/21

$$\log {}_3(x+1)=\log {}_4(x+8)$$$$x=?$$

Answered by bobhans last updated on 02/Jul/21

$$\mathrm{x}=8\Rightarrow \log {}_3(8+1)=\log {}_4(8+8)$$$$\Rightarrow 2=2$$

Commented by bobhans last updated on 02/Jul/21

$$\mathrm{let}\log {}_3(\mathrm{x}+1)=\mathrm{k}\Rightarrow \mathrm{x}+1=3^{\mathrm{k}}$$$$\mathrm{and}\log {}_4(\mathrm{x}+8)=\mathrm{k}\Rightarrow \mathrm{x}+8=4^{\mathrm{k}}$$$$\Rightarrow 4^{\mathrm{k}}-3^{\mathrm{k}}=7$$$$\mathrm{k}=0\Rightarrow 1-1\ne 7$$$$\mathrm{k}=2\Rightarrow 16-9=7\mathrm{then}\mathrm{x}=3^2-1=8\mathrm{or}$$$$\mathrm{x}=4^{\mathrm{k}}-8=16-8=8$$

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