Question 61154
I need helping solving this for x:
(ln x)^3=ln x^4
Can I rewrite it as (ln x)^3=4lnx ?


<font color = "blue">Yes.</font>


Then can I divide both sides by lnx leaving (ln x)^2=4 ?


<font color = "blue">Yes, but this assumes that you're not dividing both sides by 0, i.e. that ln(x) is not 0.  This step wouldn't be valid in the case that ln(x) = 0, so this case (ln(x) = 0) needs to be treated separately.</font>


Can I now square (root) both sides leaving me with ln x=2 ?


<font color = "blue">Right, ln (x) = 2, and also ln(x) = -2.  (Taking the square root of both sides gives ln x = +2 and ln x = -2.)  </font>


Now I'm not sure what to do next


<font color = "blue">To solve ln(x) = 2 for x, exponentiate both sides:

{{{e^(ln(x)) = e^2}}}.


The left side simplifies, giving one of the final answers:


{{{x = e^2}}}.


The equation ln (x) = -2 can be solved similarly.


Finally, there is the case when ln(x) = 0.  This happens when x = 1.  To verify that this is a solution, you can plug it into the original equation and see if it checks out (gives a true equation).</font>