Solve for x: 3^(4x-3)*5^(7x-4)=3^3x*5^(8x-7).

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to solve for x given the equation: 3^(4x-3)*5^(7x-4)=3^3x*5^(8x-7)

3^(4x-3)*5^(7x-4)=3^3x*5^(8x-7)

=> 3^4x*3^-3*5^7x*5^-4 = 3^3x*5^8x*5^-7

=> (3^4x/3^3x)*(3^-3)*(5^7x/5^8x)*(5^-4/5^-7) = 1

=> (3^x)*(3^-3)*(1/5^x)*(5^3) = 1

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

As the base is equal we equate the exponent.

=> x= 3

The required value of x is x = 3.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll divide both sides by 5^(7x-4) and 3^3x:

3^(4x-3)/3^3x = 5^(8x-7)/5^(7x-4)

We'll subtract the exponents: 3^(4x - 3 - 3x) = 5^(8x - 7 - 7x + 4)

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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