# x = ?Determine x for log 3 9 - log 3 2 = log 3 x

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### 2 Answers

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.