SOLUTION: Another one I can't figure out. I have searched the whole chapter in my book and none are quite like this. Thanks. ln(x+5)-ln(x-2)=ln(x+1)+ln(x-4)

Question 195567: Another one I can't figure out. I have searched the whole chapter in my book and none are quite like this. Thanks.
ln(x+5)-ln(x-2)=ln(x+1)+ln(x-4)
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation.

Combine the logs using the identity

Combine the logs using the identity

Since the logs have equal bases, this means that the arguments are equal.

In other words, if , then

Multiply both sides by .

FOIL

Rearrange the terms.

Expand

Distribute

Get everything to the right side.

Combine like terms.

Factor

or Set each factor equal to zero

Let's solve the first equation :

Start with the given equation.

Add 1 to both sides.

So the first possible solution is . However, if you plug this solution in the original equation, you'll end up taking the log of a negative number (which you can't do). So is NOT a solution

--------------------------------

Now let's solve :

Start with the given equation.

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

Let's use the quadratic formula to solve for x

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Rewrite as

Add to to get

Multiply and to get .

Simplify the square root (note: If you need help with simplifying square roots, check out this solver)

Break up the fraction.

Reduce.

or Break up the expression.

So the next possible solutions are or

But if you plug in , you'll once again evaluate the log of a negative number. So is NOT a solution

==================================================================
Answer:

So the only solution is
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