# SOLUTION: What process would I go through to find x in the following equation? (log = logarithm) log(1 + x) + log(2 + x) = 2 I have done logarithm problems before but I never really u

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Question 466404: What process would I go through to find x in the following equation?
(log = logarithm)
log(1 + x) + log(2 + x) = 2
I have done logarithm problems before but I never really understood the way logarithm works, I just go through the sequence my book shows me to, but I have never seen a logarithm equation such as this one before. Any help will be appreciated.

Found 3 solutions by nerdybill, solver91311, Theo:
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You have to apply the "rules for logarithms"
log(1 + x) + log(2 + x) = 2
we can rewrite the above as:
log(1 + x)(2 + x) = 2
(1 + x)(2 + x) = 10^2
expanding left side by FOILing:
2+3x+x^2 = 100
x^2+3x+2 = 100
x^2+3x-98 = 0
since the above can't be factored, you must apply the "quadratic formula" to get:
x = {8.51, -11.51}
You can throw out the negative solution (extraneous) leaving:
x = 8.51
.
 Solved by pluggable solver: SOLVE quadratic equation with variable Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=401 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 8.51249219725039, -11.5124921972504. Here's your graph:

You can put this solution on YOUR website!

The sum of the logs is the log of the product.

Unspecified base implies base 10

Definition of the logarithm function.

Not factorable. Use the quadratic formula to solve. Note: The discriminant is prime. Exclude the negative root because anything less than -1 is excluded from the domain of your most restrictive log argument, i.e., (1 + x).

Use a calculator to check your work using numerical methods.

John

My calculator said it, I believe it, that settles it

You can put this solution on YOUR website!
this is a pain in the butt because you don't get a clean answer, but i confirmed that the answer is correct, so i must be doing something right.
here's how you would solve this problem.
log(1 + x) + log(2 + x) = 2
the concepts you will use to solve this are:
concept number 1:
log(a) + log(b) = log(a * b)
concept number 2:
y = log(x) if and only if x = 10^y
note that log(x) could also be written as log(10,x) which means log of x to the base 10.
if you are dealing with the base of 10, then you don't need to show the 10 which is why log(10,x) can be shown as log(x).
if it's any other base, you would need to show the base.
for example:
log of 20 to the base of 2 would be shown as log(2,20).
the general form of this statement would be log(b,x) means log of x to the base b.
y = log(x) if and only if x = 10^y would also be written as:
y = log(10,x) if and only if x = 10^y.
the general form of this concept would be:
y = log(b,x) if and only if x = b^y
the b represents any base.
you started with the equation:
log(1 + x) + log(2 + x) = 2
using concept number 1, you transform this equation to:
log( (1 + x) * (2 + x) ) = 2
using concept number 2, you transform this equation to:
2 = log( (1 + x) * (2 + x) ) if and only if 10^2 = (1+ x) * (2 + x)
this is a basic quadratic equation.
multiply out the factors on the right side of this equation and you get:
10^2 = x^2 + 3x + 2
this is equivalent to:
x^2 + 3x + 2 = 100
subtract 100 from both sides of this equation to get:
x^2 + 3x - 98 = 0
this can't be factored by eye, so you need to resort to the quadratic formula in order to solve this quadratic equation.
x = ( (-b) +/- sqrt(b^2 - 4ac) ) / (2a)
x^2 + 3x - 98 = 0
this is in standard form of ax^2 + bx + c = 0
this means that:
a = 1
b = 3
c = -98
substituting in the quadratic formula gets us the following:
x = 8.512492197
or:
x = -11.5124922