Use logarithms to evaluate: 3^(x+2) − 3^(x−1) / 3^x and simplify your answer.

Question

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Here is the solution to the question: 3. Evaluate and simplify: (3^((x+2)) − 3^((x−1)))/(3^x) Step 1: Apply the exponent rule a^(m+n) = a^m · aⁿ and a^(m−n) = a^m · a⁻ⁿ to the terms in the numerator. 3^((x+2)) = 3^x · 3²; 3^((x−1)) = 3^x · 3⁻¹ Step 2: Substitute these expanded terms back into the expression. (3^x · 3² − 3^x · 3⁻¹)/(3^x) Step 3: Factor out the common term 3^x from the numerator. (3^x (3² − 3⁻¹))/(3^x) Step 4: Cancel out 3^x from the numerator and the denominator. 3² − 3⁻¹ Step 5: Evaluate the powers of 3. 3² = 9; 3⁻¹ = 1/3 Step 6: Perform the subtraction. To do this, find a common denominator. 9 − 1/3 = 27/3 − 1/3; (27 − 1)/(3) = 26/3 The simplified expression is 26/3.

Use logarithms to evaluate: 3^(x+2) - 3^(x-1) / 3^x and simplify your answer.

Answer

Step-by-step solution

Apply the exponent rule $a^{m+n} = a^m \cdot a^n$ and $a^{m-n} = a^m \cdot a^{-n}$ to the terms in the numerator.

Substitute these expanded terms back into the expression.

Factor out the common term $3^x$ from the numerator.

Cancel out $3^x$ from the numerator and the denominator.

Evaluate the powers of 3.

Perform the subtraction.

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