Solve for x : log 3 9 - log 3 2 = log 3 x .

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

 

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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