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December 6th, 2008, 07:29 PM   #1
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solving for x in logarithmic equations.

Hello, I have these set of questions given, I would like to know how to solve the equations (for x), and the process.

for instnace, 2logx=3log4 , where as the base of both logarithims is generally 10 in this case, I know x=8 but how do we find this, then it gets tricker for questions like this..

logx+log3= log12
log (base 2) (x+2) + log(base 2) x = 3
log (base 2) (x-2) + log (base 2 ) (x+1) = 2

If log (base 2) (log base 2) (a) = 2, what is a


For the first 2, do we change them into exponential form first, I can work my way through a) and b) but im not sure for the rest, and I don't seem to find a quick efficient way for the first 2.


Any Rules I could apply to, 2logx=3log4, that work universally for the other 4 questions too?
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December 7th, 2008, 03:48 AM   #2
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I'll assume you're interested only in real solutions.
I'll use "iff" to mean "if and only if" and "lg" to mean "log (base 2)".

2log(x) = 3log(4) iff log(x^2) = log(4^3) and x > 0, so x = (4^3)^(1/2) = 8 is the only real solution.

log(x) + log(3) = log(12) iff log(3x) = log(12), so x = 12/3 = 4 is the only real solution.

lg(x+2) + lg(x) = 3 = lg( iff lg(x(x + 2)) = lg( and x > 0, i.e., x^2 + 2x + 1 = 9 and x > 0, so x = 9^(1/2) - 1 = 2 is the only real solution.

lg(x - 2) + lg(x+1) = 2 = lg(4) iff lg((x - 2)(x + 1)) = lg(4) and x > 2, i.e., x^2 - x + 1/4 = 6 1/4 and x > 2, so x = 3 is the only real solution.

lg(lg(a)) = 2 = lg(4) iff lg(a) = 4 iff a = 2^4 = 16.
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December 7th, 2008, 12:48 PM   #3
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Re: solving for x in logarithmic equations.

Use the laws of logarithms.

For example:

Should be very usable.
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December 7th, 2008, 04:21 PM   #4
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Re: solving for x in logarithmic equations.

ooops...
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