SOLUTION: Can you please show me how to do the following problem: Find all real solutions to the equations a) ln^2x-5ln^x=-6 and b) 2 * ln(x)= ln(36)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Can you please show me how to do the following problem: Find all real solutions to the equations a) ln^2x-5ln^x=-6 and b) 2 * ln(x)= ln(36)      Log On


   

Question 316702: Can you please show me how to do the following problem: Find all real solutions to the equations a) ln^2x-5ln^x=-6
and
b) 2 * ln(x)= ln(36)
thank you!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
We're going to use the following identities/properties:


1) y%2Aln%28x%29=ln%28x%5Ey%29

2) ln%28A%29-ln%28B%29=ln%28A%2FB%29

3) If ln%28x%29=ln%28y%29, then x=y



a)

ln%282x%29-5%2Aln%28x%29=-6 Start with the given equation.


ln%282x%29-ln%28x%5E5%29=-6 Use identity #1


ln%28%282x%29%2F%28x%5E5%29%29=-6 Use identity #2


%282x%29%2F%28x%5E5%29=e%5E%28-6%29 Convert the equation into exponential form.


%282x%29%2F%28x%5E5%29=1%2Fe%5E6 Rewrite e%5E%28-6%29 as 1%2Fe%5E6


2%2F%28x%5E4%29=1%2Fe%5E6 Reduce the expression on the left side.


2=%281%2Fe%5E6%29x%5E4 Multiply both sides by x%5E4.


2e%5E6=x%5E4 Multiply both sides by e%5E6.


x%5E4=2e%5E6 Flip the equation.


x=root%284%2C2e%5E6%29 Take the fourth root of both sides.


So the solution is x=root%284%2C2e%5E6%29

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b)

2%2Aln%28x%29=ln%2836%29 Start with the given equation.


ln%28x%5E2%29=ln%2836%29 Use identity #1


x%5E2=36 Use property #3


x=sqrt%2836%29 Take the square root of both sides (note: we're ignoring the negative square root since you can't take the log of a negative number).


x=6 Take the square root of 36 to get 6.


So the answer is x=6