Question and Answers Forum

All Questions      Topic List

Logarithms Questions

Previous in All Question      Next in All Question      

Previous in Logarithms      Next in Logarithms      

Question Number 48202 by F_Nongue last updated on 20/Nov/18

$$howcanIgetthex?$$$$a)3^x+4^x=5^x$$$$b)7^{6-x}=x+2$$$$c){\left(\sqrt{2+\sqrt{3}}\right)}^x+{\left(\sqrt{2-\sqrt{3}}\right)}^x=2^x$$$$d)3^{x-2}=\frac{9}{x}$$$$$$

Commented by F_Nongue last updated on 20/Nov/18

Please help

Answered by MJS last updated on 21/Nov/18

$$\mathrm{first},\mathrm{try}\mathrm{to}\mathrm{find}\mathrm{obvious}\mathrm{solutions}$$$$\left(a\right)x=2\left[\mathrm{because}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{a}\mathrm{Pythagorean}\mathrm{triple}\right]$$$$\left(b\right)x=5[\mathrm{trying}x+2=7^n]$$$$\left(c\right)x=2\left[\mathrm{to}\mathrm{get}\mathrm{rid}\mathrm{of}\sqrt{3}\right]$$$$\left(d\right)x=3[\mathrm{because}{3}^1=\frac{9}{3}]$$$$\mathrm{second},\mathrm{to}\mathrm{find}\mathrm{if}\mathrm{there}\mathrm{are}\mathrm{more}\mathrm{solution},$$$$\mathrm{differentiate}\mathrm{and}\mathrm{approximate}\mathrm{for}\mathrm{tbe}\mathrm{valued}$$$$\mathrm{close}\mathrm{to}\mathrm{the}\mathrm{solutions}$$$$\left(a\right)f\left(x\right)=3^x+4^x-5^x\Rightarrow f^{\prime }\left(x\right)=3^x\ln 3+4^x\ln 4-5^x\ln 5$$$$\mathrm{we}\mathrm{find}{f}^{\prime }\left(x\right)\to 0\mathrm{for}x\to -\mathrm{\infty }$$$$f^{\prime }\left(x\right)\to -\mathrm{\infty }\mathrm{for}x\to +\mathrm{\infty }$$$$\mathrm{and}\mathrm{a}\mathrm{peak}\mathrm{at}x\approx 1.287\Rightarrow \mathrm{no}\mathrm{other}\mathrm{solution}$$$$\left(b\right)f\left(x\right)=7^{6-x}-(x+2)\Rightarrow f^{\prime }\left(x\right)=-(7^{6-x}\ln 7+1)$$$$\mathrm{this}\mathrm{is}\mathrm{negative}\mathrm{for}x\in \mathbb{R}\Rightarrow \mathrm{no}\mathrm{other}\mathrm{solution}$$$$\left(c\right)f\left(x\right)={(2+\sqrt{3})}^{\frac{x}{2}}+{(2-\sqrt{3})}^{\frac{x}{2}}-2^x\Rightarrow$$$$\Rightarrow f^{\prime }\left(x\right)=\frac{\ln (2+\sqrt{3})}{2}{(2+\sqrt{3})}^{\frac{x}{2}}+\frac{\ln (2-\sqrt{3})}{2}{(2+\sqrt{3})}^{\frac{x}{2}}-2^x\ln 2$$$$\mathrm{this}\mathrm{is}\mathrm{negative}\mathrm{for}x\in \mathbb{R}\Rightarrow \mathrm{no}\mathrm{other}\mathrm{solution}$$$$\left(d\right)f\left(x\right)=3^{x-2}-\frac{9}{x}\Rightarrow f^{\prime }\left(x\right)=3^{x-2}\ln 3+\frac{9}{x^2}$$$$\mathrm{this}\mathrm{is}\mathrm{positive}\mathrm{for}x\in \mathbb{R}\mathrm{\setminus }\left\{0\right\}\Rightarrow \mathrm{no}\mathrm{other}\mathrm{solution}$$

Commented by Cheyboy last updated on 21/Nov/18

$$\mathrm{Nice}\mathrm{Sir}$$

Commented by F_Nongue last updated on 21/Nov/18

thanks sir.

Terms of Service

Privacy Policy

Contact: info@tinkutara.com