Exercise 12 Page 303

To solve the given exponential equation, we will start by rewriting the terms so that they have a common base. Note that 27 cannot be represented as a natural power of 9. Therefore, we will write both 27 and 9 as powers of 3. Now, we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 3^(3x) = 3^(2x-4) ⇔ 3x = 2x-4 Finally, we will solve the equation 3x = 2x-4 by subtracting 2x from both sides of the equation. 3x = 2x-4 ⇔ x = -4 To check our answer, we will substitute -4 for x in the given equation.

27^x = 9^(x-2)

▼

x= -4

27^{{\color{#0000FF}{\text{-}4}}} \stackrel{?}{=} 9^{{\color{#0000FF}{\text{-}4}}-2}

Subtract term

27^{\text{-}4} \stackrel{?}{=} 9^{\text{-}6}

Write as a power

\left( 3^3 \right)^{\text{-}4} \stackrel{?}{=} \left( 3^2 \right)^{\text{-}6}

'"`UNIQ--MLMath-9-QINU`"'

3^{3(\text{-}4)} \stackrel{?}{=} 3^{2(\text{-}6)}

'"`UNIQ--MLMath-11-QINU`"'

3^(-12) = 3^(-12) ✓

Since substituting -4 for x in the given equation produces a true statement, x=-4 is the solution to our equation.