Question 624523: solve the equation 9 log5 x = 25 logx 5, expressing your answers in the form 5^p/q, where p, and q are integers
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! original equation to be solved is:
9*log(5,x) = 25*log(x,5)
log(5,x) means log of x to the base of 5.
log(x,5) means log of 5 to the base of x.
convert everything to base of 5 using the base conversion formula of log(b,x) = log(c,x)/log(c,b).
original equation of 9*log(5,x) = 25*log(x,5) becomes:
9*log(5,x) = 25*log(5,5)/log(5,x)
since log(5,5) is equal to 1, this equation becomes:
9*log(5,x) = 25/log(5,x)
multiply both sides of this equation by log(5,x) and divide both sides of this equation by 9 to get:
log(5,x) * log(5,x) = 25/9
this can be written as:
(log(5,x))^2 = 25/9
take the square root of both sides of this equation to get:
log(5,x) = +/- sqrt(25/9) which becomes:
log(5,x) = +/- 5/3
your possible answers are:
log(5,x) = 5/3
log(5,x) = -5/3
from the basic definition of logarithms, you get:
log(b,x) = c if and only if b^c = x
apply this to your first possible answer and you get:
log(5,x) = 5/3 if and only if 5^(5/3) = x
that's one of your possible answers.
x = 5^(5/3)
apply this to your second possible answer and you get:
log(5,x) = -5/3 if and only if 5^(-5/3) = x
that's the other of your possible answers.
5^(5/3) resolves to 14.62008869
5^-(5/3) resolves to 1/5^(5/3) which resolves to .068399038
to see if these answers are good, you need to substitute them into your original equation by substituting them in place of x.
before you do that, however, you need to convert equation to base of 10 so you can solve it using your calculator.
your original equation of:
9*log(5,x) = 25*log(x,5) becomes:
9*LOG(x)/LOG(5) = 25*LOG(5)/LOG(x)
all you need to do now is substitute.
when x = 14.62008869, your formula becomes:
9*LOG(14.6200869)/LOG(5) = 25*LOG(5)/LOG(14.6200869)
solve this using your calculator and you get:
15 = 15, so x = 14.62008869 is good.
remember 14.62008869 is equivalent to 5^(5/3)
when x = .068399038, your formula becomes:
9*LOG(.068399038)/LOG(5) = 25*LOG(5)/LOG(.068399038)
solve this using your calculator and you get:
-15 = -15
remember .068399038 is equivalent to 5^-(5/3) which is equialent to 1/5^(5/3)
both answers are good.
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