Question 651641: Solve for x if log10(3x+1) + log10(1/2) -log10(2x-5)=0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your problem is:
log(3x+1) + log(1/2) - log(2x-5) = 0
note that log(x) is equivalent to log(10,x) which is equivalent to the log of x to the base of 10.
note also that the log function of your calculator will solve for the log of x to the base of 10 which would be shown as LOG(x).
once again, your problem is:
log(3x+1) + log(1/2) - log(2x-5) = 0
couple of properties of logarithms that will help you out.
log(x*y) = log(x) + log(y)
log(x/y) = log(x) - log(y)
once again, your problem is:
log(3x+1) + log(1/2) - log(2x-5) = 0
using the first property, you get:
log(3x+1) + log(1/2) is equivalent to:
log((3x+1)*(1/2))
your problem now becomes:
log((3x+1)*(1/2)) - log(2x-5) = 0
using the second property, you get:
log((3x+1)*(1/2)) - log(2x-5) is equivalent to:
log((3x+1)*(1/2)/(2x-5))
your problem now becomes:
log((3x+1)*(1/2)/(2x-5)) = 0
simplify the expression within the logarithm sign to get:
(3x+1)*(1/2)/(2x-5) is equivalent to:
(3x+1)/(2*(2x-5)) which is equivalent to:
(3x+1) / (4x-10)
your problem now becomes:
log((3x+1)/(4x-10)) = 0
the law of logarithms states that:
y = log(b,x) if and only if b^y = x
since the base is 10, applying this law to your problem provides the following:
log((3x+1)/(4x-10)) = 0 if and only if 10^0 = (3x+1)/(4x-10)
since 10^0 is equal to 1, your problem becomes:
1 = (3x+1)/(4x-10)
multiply both sides of this equation by (4x-10) to get:
4x-10 = 3x+1
subtract 3x from both sides of this equation and add 10 to both sides of this equation to get:
x = 11
that's you solution.
to confirm this solution is good, substitute for x in your original equation of:
log(3x+1) + log(1/2) - log(2x-5) = 0 to get:
log(3(11)+1) + log(1/2) - log(2(11)-5) = 0 which simplifies to:
log(34) + log(1/2) - log(17) = 0
use your calculator to get:
LOG(35) + LOG(1/2) - LOG(17) = 0
this results in 0 = 0 which is true, confirming the value of 11 for x is good.
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