Question 558882: Solve the logarithmic equation. Express your solutions in exact form only. Please show all of your work.
ln(3x+5)+ ln(3x-5)= 6 +ln4
THANK YOU!!!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is:
ln(3x+5) + ln(3x-5) = 6 + ln(4)
subtract ln(4) from both sides of the equation to get:
ln(3x+5) + ln(3x-5) - ln(4) = 6
since ln(a) + ln(b) = ln(a*b), this equation becomes:
ln((3x+5)*(3x-5)) - ln(4) = 6
since ln(a) - ln(b) = ln(a/b), this equation becomes:
ln(((3x+5)*(3x-5))/4) = 6
since ln(a) = b if and only if e^5 = a, then you get:
e^6 = ((3x+5)*(3x-5))/4
multiply both sides of this equation by 4 to get:
4e^6 = (3x+5) * (3x-5)
since (3x+5) * (3x-5) = 9x^2 - 25, this equaton becomes:
4e^6 = 9x^2 - 25
add 25 to both sides of this equation to get:
9x^2 = 4e^6 + 25
divide both sides of this equation by 9 to get:
x^2 = (4e^6 + 25)/9
take the square root of both sides of this equation to get:
x = +/- sqrt((4e^6 + 25)/9)
that's your answer.
you can use your calculator to solve for the value of x.
you should get:
x = +/- 13.49368237
you can then substitute for x in your original equation and the equation should hold true.
i've done it and it does so i'm convinced.
actually, only x = + 13.49368237 works.
x = - 13.49368237 doesn't because then you are taking natural log of a negative number which is invalid.
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