Tatiana Cook

2022-10-08

I can't find second solution to this logarithmic problem!

I kind of got stuck on one step in solving a logarithmic equation.

The equation given was: x^3lnx - 4xlnx = 0

My steps so far:

x^3lnx - 4xlnx = 0

ln((x^x^3)/(x^4x)) = 0

e^ln((x^x^3)/(x^4x)) = e^0

(x^x^3)/(x^4x) = 1

x^x^3 = x^4x

now I would just remove the base to make it x^3=4x. However, this step would also remove the solution x = 1 from the equation. I only got it through guessing and then checking on a graphic calculator.

The second asnwer, x = -2 I do know how to get. I just solved

x^3 - 4x = 0

x(x^2 - 4) = 0

x(x-2)(x+2) = 0 (so the solutions could be +/- 2. By plugging these values back in the original formula I found out that only -2 is the solution.)

THE QUESTION: Can someone please show me how to algebraically find the solution x = 1?

I kind of got stuck on one step in solving a logarithmic equation.

The equation given was: x^3lnx - 4xlnx = 0

My steps so far:

x^3lnx - 4xlnx = 0

ln((x^x^3)/(x^4x)) = 0

e^ln((x^x^3)/(x^4x)) = e^0

(x^x^3)/(x^4x) = 1

x^x^3 = x^4x

now I would just remove the base to make it x^3=4x. However, this step would also remove the solution x = 1 from the equation. I only got it through guessing and then checking on a graphic calculator.

The second asnwer, x = -2 I do know how to get. I just solved

x^3 - 4x = 0

x(x^2 - 4) = 0

x(x-2)(x+2) = 0 (so the solutions could be +/- 2. By plugging these values back in the original formula I found out that only -2 is the solution.)

THE QUESTION: Can someone please show me how to algebraically find the solution x = 1?

Emmalee Reilly

Beginner2022-10-09Added 6 answers

$\begin{array}{rl}{x}^{3}\mathrm{ln}x-4x\mathrm{ln}x=0& \phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}({x}^{3}-4x)\mathrm{ln}x=0\\ \\ & \phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}x({x}^{2}-4)\mathrm{ln}x=0\\ \\ & \phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}x(x-2)(x+2)\mathrm{ln}x=0\end{array}$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}x=0\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}x-2=0\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}x+2=0\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\mathrm{ln}x=0,$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}x=0\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x=2\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x=-2\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x=1$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}x=0\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}x-2=0\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}x+2=0\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\mathrm{ln}x=0,$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}x=0\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x=2\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x=-2\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\text{or}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}x=1$

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