Exercise 19 Page 303

Finding the Error

Let's start by determining where the error occurred. Let's look at the first step. 5^(3x+2)&=25^(x-8) &⇓ * 3x+2&=x-8 The error occurred when going from the first line to the second line; there is an incorrect application of the Property of Equality for Exponential Equations.

b^x&=b^y &⇓ x&=y This property can only be applied whenever the bases are the same, and in our case the bases were not.

Correctly Solving the Equation

In order to correctly solve this equation, we need to make sure that the bases are equal. To do this we will rewrite 25 as 5^2 and simplify.

5^(3x+2)=25^(x-8)

Rewrite

Rewrite 25 as 5^2

5^(3x+2)=(5^2)^(x-8)

PowPow

(a^m)^n=a^(m* n)

5^(3x+2)=5^(2x-16)

Now we can apply the Property of Equality for Exponential Equations and solve for x.

5^(3x+2)=5^(2x-16)

EquateExponents

Equate exponents

3x+2=2x-16

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Solve for x

SubEqn

LHS-2x=RHS-2x

1x+2=- 16

IdPropMult

Identity Property of Multiplication

x+2=- 16

SubEqn

LHS-2=RHS-2

x=- 18

When solved correctly, x=- 18.