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Question Number 53912 by Mikael_Marshall last updated on 27/Jan/19

$$\{\begin{array}{l}{2}^x-2^y=1\\ 4^x-4^y=\frac{5}{3}\end{array}$$

Answered by byaw last updated on 27/Jan/19

$$2^x=1+2^y$$$$2^{2x}={(1+2^y)}^2$$$$4^x=1+2.2^y+4^y$$$$4^x-4^y=1+2.2^y$$$$1+2.2^y=5/3$$$$3+6.2^y=5$$$$6.2^y=2$$$$2^y=1/3$$$$y=\log {}_2\left(1/3\right)$$$$2^x-2^{\log {}_2\left(1/3\right)}=1$$$$2^x-1/3=1$$$$2^x=4/3$$$$x=\log {}_2\left(4/3\right)$$$$x=2-\log {}_2\left(1/3\right)$$

Commented by Mikael_Marshall last updated on 27/Jan/19

$$thanksSir!$$

Commented by JDamian last updated on 27/Jan/19

$$$$$$\mathrm{Hi},\mathbf{byaw}.\mathrm{Your}\mathrm{last}\mathrm{line}\mathrm{is}\mathrm{wrong}.\mathrm{It}\mathrm{should}\mathrm{be}$$$$x=2-{\log }_{2}3$$

Commented by byaw last updated on 15/Feb/19

$$yesIverealized$$

Answered by peter frank last updated on 27/Jan/19

$$a-b=1$$$$a^2-b^2=\frac{5}{3}$$$$(a-b)(a+b)=\frac{5}{3}$$$$a+b=\frac{5}{3}$$$$2a=\frac{5}{3}+1=\frac{8}{3}$$$$a=\frac{8}{6}$$$$-2b=1-\frac{5}{3}$$$$2b=\frac{5}{3}-1$$$$b=\frac{2}{6}$$$$$$

Commented by Mikael_Marshall last updated on 27/Jan/19

$$thanksSir$$

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Jan/19

$$a-b=1$$$$a^2-b^2=\frac{5}{3}$$$$a+b=\frac{5}{3}$$$$a-b=1$$$$2a=1+\frac{5}{3}a=\frac{4}{3}$$$$b=\frac{5}{3}-\frac{4}{3}=\frac{1}{3}$$$$2^x=\frac{4}{3}$$$$x={ln}_2\left(\frac{4}{3}\right)$$$$2^y=\frac{1}{3}y={ln}_2\left(\frac{1}{3}\right)$$

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