Question 1074900: Solve log8(5)-log8(2x)=log8(9)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! start with log8(5) - log8(2x) = log8(9)
subtract log8(5) from both sides to get -log8(2x) = log8(9) - log8(5)
multiply both sides of the equation by -1 to get log8(2x) = log8(5) - log8(9)
since log(x) - log(y) = log(x/y), you get log8(2x) = log8(5/9)
this is true if and only if 2x = 5/9.
solve for x to get x = 5/18.
confirm by replacing x with 5/18 in the original equation.
you get log8(5) - log8(2*5/18) = log8(9)
simplify to get log8(5) - log8(5/9) = log8(9)
since log(x) - log(y) = log(x/y), this becomes log8(5/(5/9)) = log8(9)
simplify to get log8(5*9/5) = log8(9)
simplify further to get log8(9) = log8(9).
you can also confirm by using the log base conversion formula of:
log8(x) = log(x)/log(8)
log(x) means log10(x) which is the LOG function of your calculator.
your original equation is:
log8(5) - log8(2x) = log8(9)
use the base conversion formula to get log(5)/log(8) - log(2x)/log(8) = log(9)/log(8)
multiply both sides of the equation by log(8) to get log(5) - log(2x) = log(9).
now you're in base 10 and you can use your calculator to confirm the solution.
your solution was x = 5/18 which made 2x = 5/9.
replace 2x with 5/9 and your equation in base 10 becomes log(5) - log(5/9) = log(9)
use your calculator to confirm.
you get:
log(5) - log(5/9) = .9542425094
log(9) = the same.
it didn't matter whether you were in base 10 or base 8.
this equation was valid regardless of the base of the log.
the reason is because log(x) = log(y) if and only if x = y.
this fact was shown above in the statements:
log8(2x) = log8(5/9)
this is true if and only if 2x = 5/9.
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