# What is x for log 3 9 - log 3 2 = log 3 x?

*print*Print*list*Cite

### 3 Answers

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.