Solve: logx + log(x + 15) = 2?

Question

How do you solve the equation log(x) + log(x + 15) = 2? I understand it's the logarithm of x plus the logarithm of (x + 15) equaling 2, but I'm not sure how to approach solving it. Could someone explain the steps?

Anonymous · Accepted Answer

You need to use the logarithm identity,
log(A*B) = log(A) + log(B)
so,
logx + log(x+15) = 2
log(x*(x+15)) = 2
log (x^2 + 15x) = 2
I am assuming you mean log base 10. In that case, we have:
x^2 + 15x = 10^2
x^2 + 15x = 100
x^2 + 15x - 100 = 0
Factoring,
(x + 20)(x - 5) = 0
So, x = -20 or x = 5.

Prof. Shane Goldner · Answer

log A + log B = log (A • B)
logx + log(x + 15) = 2
log (x • (x + 15) ) = 2
log (x² + 15x) = 2
10² = x² + 15x
100 = x² + 15x
x² + 15x - 100 = 0
(x + 20) (x - 5) = 0
x = -20 and x = 5
Plugging them both into the equation you will find that only x = 5 will work since you can't take the log of a negative number so x = -20 is an extraneous solution. 
The only answer is x = 5.

Anonymous · Answer

logx + log(x + 15) = 2
log(x^2+15x) = 2
100 = x^2 + 15x
x^2 + 15x - 100 = 0
x^2 + 20x - 5x - 100 = 0
x(x+20) - 5(x+20) = 0
(x-5)(x+20) = 0
x = 5 or -20
x = 5

Dr. Otto Hansen DDS · Answer

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this is the same as writing: log(x(x-15)) = 2 log(x^2 - 15) = 2 if the base for log is not written then you assume that it is to the base 10 therefore 10^2 = x^2 - 15 100 + 15 = x^2 x^2 = 115 therefore x = sqareroot(115) type that into your calculator and that should give you the answer ? good luck!

Anonymous · Answer

us your log rules to combine the logs so it becomes log x squared +15x =2  then use the inverser so its 10 squared=x squared+ 15x and factor and if you cant factor use quadratic formula but just a hint you can factor it ?

Alvina Sanford

Solve: logx + log(x + 15) = 2?

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