Answer to Question #171294 in Algebra for gibson

Question #171294

solve for x in the following equations

  1. 4x-2 = 3x+4
  2. 9x-3x-8=0
1
Expert's answer
2021-03-16T07:14:09-0400

1) Let's take logarithm of both sides of equation:


log(4x2)=log(3x+4),log(4^{x-2})=log(3^{x+4}),(x2)log(4)=(x+4)log(3),(x-2)\cdot log(4)=(x+4)\cdot log(3),xlog(4)2log(4)=xlog(3)+4log(3),xlog(4)-2log(4)=xlog(3)+4log(3),xlog(4)xlog(3)=4log(3)+2log(4),xlog(4)-xlog(3)=4log(3)+2log(4),x(log(4)log(3))=4log(3)+2log(4),x(log(4)-log(3))=4log(3)+2log(4),x(log(43))=log(81)+log(16),x(log(\dfrac{4}{3}))=log(81)+log(16),x(log(43))=log(1296),x(log(\dfrac{4}{3}))=log(1296),x=log(1296)log(43).x=\dfrac{log(1296)}{log(\dfrac{4}{3})}.

Answer:

x=log(1296)log(43).x=\dfrac{log(1296)}{log(\dfrac{4}{3})}.

2)

9x3x8=0,9^x-3^x-8=0,32x3x8=0.3^{2x}-3^x-8=0.

Let 3x=y3^x=y, then we get the following quadratic equation:


y2y8=0.y^2-y-8=0.

This quadratic equation has two roots:


y1=b+b24ac2a=1+332,y_1=\dfrac{-b+\sqrt{b^2-4ac}}{2a}=\dfrac{1+\sqrt{33}}{2},y2=bb24ac2a=1332.y_2=\dfrac{-b-\sqrt{b^2-4ac}}{2a}=\dfrac{1-\sqrt{33}}{2}.

Let's go back, substitute yy and solve for xx:


3x=1+332,3^x=\dfrac{1+\sqrt{33}}{2},log(3x)=log(1+332),log(3^x)=log(\dfrac{1+\sqrt{33}}{2}),xlog(3)=log(1+332),xlog(3)=log(\dfrac{1+\sqrt{33}}{2}),x=log(1+332)log(3).x=\dfrac{log(\dfrac{1+\sqrt{33}}{2})}{log(3)}.

There is no solution for case when we substitute y2=1332.y_2=\dfrac{1-\sqrt{33}}{2}.

Answer:

x=log(1+332)log(3).x=\dfrac{log(\dfrac{1+\sqrt{33}}{2})}{log(3)}.


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