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:
Answer by nerdybill(7384)   (Show Source):
Answer by solver91311(24713)  (Show Source):
Answer by Theo(13342)  (Show Source):
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.
you start with:
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.
under concept number 2, please be advised of the following:
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.
going back to your problem:
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)
solve this equation and you have your answer.
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.
the quadratic formula is:
x = ( (-b) +/- sqrt(b^2 - 4ac) ) / (2a)
your quadratic equation is:
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
those are your answers.
if you substitute either of those values for x in your original equation, you will see that the equations are true, confirming these answers are good.
for example:
using the 8.5 number, your original equation of:
log(1 + x) + log(2 + x) = 2 becomes:
log(1 + 8.512492197) + log(2 + 8.512492197) = 2 which becomes:
log(9.512492197) + log(10.512492197) = 2 which becomes:
.978294314 + 1.021705686 = 2, confirming that the equation is true when the value of x is replaced with 8.512492197.
note that i used my calculator to get the log of 9.512492197 and to get the log of 10.512492197.
my statement about this being a pain in the butt is because the factorization of the quadratic equation wasn't clean and we had to deal with fractional numbers requiring the use of the calculator to find the answer. I stored intermediate results in memory so I didn't have to type all the decimal places that i showed you here.
so, that's how it's done.
the basic concepts i showed you are what you need to use to solve this, plus you need to know the quadratic formula to solve for the quadratic equation.
plus you needed to recognize that you had a quadratic equation in the first place.
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