15^(2x-3)=3^x*5^(3x-6) x=?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to solve for x given: 15^(2x-3)=3^x*5^(3x-6)

15^(2x-3)=3^x*5^(3x-6)

=> (5*3)^(2x - 3) = 3^x * 5^(3x - 6)

=> 5^(2x - 3) * 3^(2x - 3) = 3^x * 5^(2x - 3)* 5^x / 5^3

=> 3^(2x - 3) = 3^x * 5^x / 5^3

=> 125* 3^2x / 3^3 = 3^x * 5^x

=> 125 * 3^x * 3^x = 27 * 3^x * 5^x

=> 125 * 3^x = 27 * 5^x

=> 125 / 27 = 5^x / 3^x

=> (5/3)^3 = (5/3)^x

As the base is equal equate the exponent.

We get x = 3.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We notice that we can write 15 = 3*5

We'll raise to 2x-3 both sides:

15^(2x-3) = 3^(2x-3)*5^(2x-3)

We'll re-write the equation:

3^(2x-3)*5^(2x-3) = 3^x*5^(3x-6)

We'll divide by 3^x and we'll get:

3^(2x-3)*5^(2x-3)/3^x = 5^(3x-6)

We'll divide by 5^(2x-3):

3^(2x-3)/3^x = 5^(3x-6)/5^(2x-3)

We'll subtract the exponents:

3^(2x - 3 - x) = 5^(3x - 6 - 2x + 3)

We'll combine like terms inside brackets:

3^(x - 3) = 5^(x - 3)

We'll re-write the equation:

3^x*3^-3 = 5^x*5^-3

3^x/3^3 = 5^x/5^3

We'll create matching bases. We'll divide by 5^x:

3^x/5^x*3^3 = 1/5^3

We'll multiply by 3^3:

3^x/5^x = 3^3/5^3

(3/5)^x = (3/5)^3

Since the bases are matching, we'll apply one to one property:

x = 3

The solution of the equation is x = 3.

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