Simplify the logarithmic expression: 23 9 − 1/23 144 + 3 972

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Step 1: Apply the power rule of logarithms, a log_b x = log_b (x^a), to each term. 2log₃ 9 = log₃ (9²) = log₃ 81; 1/2log₃ 144 = log₃ (144^(1/2)) = log₃ √(144) = log₃ 12 The expression becomes: log₃ 81 − log₃ 12 + log₃ 972 Step 2: Apply the quotient and product rules of logarithms, log_b x − log_b y = log_b (x/y) and log_b x + log_b y = log_b (xy). Combine the terms into a single logarithm: log₃ 81 − log₃ 12 + log₃ 972 = log₃ (81/12 × 972) log₃ ((81 × 972)/(12)) Step 3: Simplify the argument of the logarithm. Divide 972 by 12: 972 ÷ 12 = 81 Substitute this back into the argument: (81 × 972)/(12) = 81 × 81 = 81² Step 4: Express the argument as a power of the base 3. We know that 81 = 3⁴. So, 81² = (3⁴)² = 3^(4 × 2) = 3⁸. Step 5: Substitute this back into the logarithm. The expression becomes: log₃ (3⁸) Step 6: Use the property log_b (bⁿ) = n. log₃ (3⁸) = 8 The final answer is 8.

Simplify the logarithmic expression: 23 9 - (1)/(2)3 144 + 3 972

Answer

Step-by-step solution

Apply the power rule of logarithms, $a \log_b x = \log_b (x^a)$ , to each term.

Apply the quotient and product rules of logarithms, $\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)$ and $\log_b x + \log_b y = \log_b (xy)$ .

Simplify the argument of the logarithm.

Express the argument as a power of the base 3.

Substitute this back into the logarithm.

Use the property $\log_b (b^n) = n$ .

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