We have to find x given that log(3) 9 - log(3) 2 = log(3) x
Use the property of logarithm, log a - log b = log (a/b)
log(3) 9 - log(3) 2 = log(3) x
=> log(3)(9/2) = log(3)x
As the log on both the sides is to the same base, we can equate x = 9/2
The required value of x = 9/2
We'll impose one constraints for the logarithm log 3 x to exist.
x > 0
We'll use the power property of logarithms:
log 3 9 = 2* log 3 3 = 2
We'll use the power property of logarithms and the symmetric property:
log 3 (x) = log 3 (3)^2 - log 3 (2)
Because the bases are matching, we'll transform the difference of logarithms from the right side, into a quotient. We'll apply the formula:
lg a - lg b = lg (a/b)
We'll substitute a by 9 and b by 2. The logarithms from formula are decimal logarithms. We notice that the base of logarithm is 3.
log 3 (x) = log 3 (9/2)
Since the bases are matching, we'll apply the one to one property:
x = 9/2
x = 4.5
Since the value of x >0, we'll validate x = 4.5 as the solution of the equation.