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 4477 by love math last updated on 30/Jan/16

2 log (x−2)+log (x−4)^2 =0 Determine the domain of x and find the value of x.

$$\mathrm{2}\:{log}\:\left({x}−\mathrm{2}\right)+{log}\:\left({x}−\mathrm{4}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$ \\ $$$${Determine}\:{the}\:{domain}\:{of}\:{x}\:{and}\:{find}\:{the}\:{value}\:{of}\:{x}. \\ $$

Answered by Rasheed Soomro last updated on 30/Jan/16

2 log (x−2)+log (x−4)^2 =0 2log(x−2)=−log(x−4)^2 log(x−2)^2 =log(x−4)^(−2) (x−2)^2 =(x−4)^(−2) (x−2)^2 =((1/(x−4)))^2 x−2=±(1/(x−4)) (x−2)(x−4)=±1 x^2 −6x+8−1=0 ∣ x^2 −6x+8+1=0 x^2 −6x+7=0 ∣ x^2 −6x+9=0 x=((−(−6)±(√((−6)^2 −4(1)(7))))/2) ∣ (x−3)^2 =0 x=((6±(√8))/2)=3±(√2) ∣ x=3 Domain of equation :{3,3±(√2)}

$$\mathrm{2}\:{log}\:\left({x}−\mathrm{2}\right)+{log}\:\left({x}−\mathrm{4}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{2}{log}\left({x}−\mathrm{2}\right)=−{log}\left({x}−\mathrm{4}\right)^{\mathrm{2}} \\ $$$${log}\left({x}−\mathrm{2}\right)^{\mathrm{2}} ={log}\left({x}−\mathrm{4}\right)^{−\mathrm{2}} \\ $$$$\left({x}−\mathrm{2}\right)^{\mathrm{2}} =\left({x}−\mathrm{4}\right)^{−\mathrm{2}} \\ $$$$\left({x}−\mathrm{2}\right)^{\mathrm{2}} =\left(\frac{\mathrm{1}}{{x}−\mathrm{4}}\right)^{\mathrm{2}} \\ $$$${x}−\mathrm{2}=\pm\frac{\mathrm{1}}{{x}−\mathrm{4}} \\ $$$$\left({x}−\mathrm{2}\right)\left({x}−\mathrm{4}\right)=\pm\mathrm{1} \\ $$$${x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{8}−\mathrm{1}=\mathrm{0}\:\:\mid\:{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{8}+\mathrm{1}=\mathrm{0} \\ $$$${x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{7}=\mathrm{0}\:\:\:\mid\:\:{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{9}=\mathrm{0} \\ $$$${x}=\frac{−\left(−\mathrm{6}\right)\pm\sqrt{\left(−\mathrm{6}\right)^{\mathrm{2}} −\mathrm{4}\left(\mathrm{1}\right)\left(\mathrm{7}\right)}}{\mathrm{2}}\:\:\mid\:\:\left({x}−\mathrm{3}\right)^{\mathrm{2}} =\mathrm{0} \\ $$$${x}=\frac{\mathrm{6}\pm\sqrt{\mathrm{8}}}{\mathrm{2}}=\mathrm{3}\pm\sqrt{\mathrm{2}}\:\:\:\:\:\:\:\mid\:\:\:{x}=\mathrm{3} \\ $$$${Domain}\:{of}\:{equation}\::\left\{\mathrm{3},\mathrm{3}\pm\sqrt{\mathrm{2}}\right\} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com