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