SOLUTION: solve the logarithm 2log3z-log3(z-2)=2

Question 174880: solve the logarithm

2log3z-log3(z-2)=2
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Take note that (for the first log) and ---> (for the second log).

So the domain is (this interval works for BOTH logs)

Rewrite the first log using the identity

Combine the logs using the identity

Rewrite the equation using the property: ====>

Square 3 to get 9

Multiply both sides by .

Distribute.

Subtract from both sides.

Notice we have a quadratic equation in the form of where , , and

Let's use the quadratic formula to solve for z

Start with the quadratic formula

Plug in , , and

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

===================================================

Answer:

So the solutions are or

Take note that both of these solutions are within the domain . So they are valid solutions.

Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!

These solutions check if I plug them back in
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