Question 1074900
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.