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Question Number 1304 by 314159 last updated on 20/Jul/15

$$If{3}^{log3x}=4^{log4x},findx.$$$$$$

Answered by Rasheed Soomro last updated on 20/Jul/15

$$3^{log3x}=4^{log4x}$$$$Takinglogarthmofbothsides$$$$log\left(3^{log3x}\right)=log\left(4^{log4x}\right)$$$$log3x.log3=log4x.log4$$$$(log3+logx).log3=(log4+logx).log4$$$${\left(log3\right)}^2+log3.logx={\left(log4\right)}^2+log4.logx$$$$log4.logx-log3.logx={\left(log3\right)}^2-{\left(log4\right)}^2$$$$logx.(log4-log3)=(log3+log4)(log3-log4)$$$$-logx(log3-log4)=(log3+log4)(log3-log4)$$$$-logx=log3+log4$$$$log\left(x^{-1}\right)=log\left(3\times 4\right)$$$$\frac{1}{x}=12$$$$x=\frac{1}{12}$$

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