What is x for log 3 9 - log 3 2 = log 3 x?
The logarithmic equation `log_3 9 - log_3 2 = log_3 x` has to be solved for x. The base of the logarithm here is 3.
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`
`log_3 4.5 = log_3 x`
Now just equate the terms of which the base is being taken on both the sides.
This gives the solution of the equation as x = 4.5. There is no need to do anything else.
We have to find x for log 3 9 - log 3 2 = log 3 x
Now for logarithms we have the relations that log (a*b) = log a + log b and log (a/b) = log a - log b.
Now log 3 9 - log 3 2 = log 3 x
=> log 3 ( 9/2) = log 3 x
raising both sides to the power 3
=> 9/2 = x
We also see that x = 9/2 is a positive number as hence satisfies the condition that logarithms can be taken only of positive numbers.
Therefore 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.