x = ?Determine x for log 3 9 - log 3 2 = log 3 x
You need to use the propreties of logarithms when solve this equation.
Using the power property of logarithms you may write the term log_3 9 = log_3 (3^2) = 2log_3 3
Adding log_3 2 both sides yields:
2 = log_3 x + log_3 2
You need to transform the sum of logarithms log_3 x + log_3 2 in logarithm of product using product rule such that:
log_3 x + log_3 2 = log_3 (2x)
Writing the new form of equation yields:
2 = log_3 (2x) => 2x = 3^2 => x = 9/2
Plugging x=9/2 in equation yields:
log 3 9 - log 3 2 = log 3 (9/2)
Using the property that the log of quotient is the difference of logs yields:
log_3 (9/2) = log_3 9 - log_3 2
Hence, using this property, both sides become equal such that:
log 3 9 - log 3 2 = log 3 9 - log 3 2
Hence, the solution to equation is x=9/2.
For the logarithm to exist, x has to be positive.
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)
Because the bases are matching, we'll apply the one to one property:
x = 9/2
x = 4.5
Since the value of x is positive, the solution of the equation is valid.