Simplify the logarithmic expression: 1/23 36 − 3 2 − 33 3

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Apply the power rule of logarithms, c log_b M = log_b (M^c), to each term. 1/2log₃ 36 = log₃ (36^(1/2)) = log₃ √(36) = log₃ 6; 3log₃ 3 = log₃ (3³) = log₃ 27 Step 2: Substitute these simplified terms back into the original expression. log₃ 6 − log₃ 2 − log₃ 27 Step 3: Apply the quotient rule of logarithms, log_b M − log_b N = log_b (M/N), from left to right. First, combine the first two terms: log₃ 6 − log₃ 2 = log₃ (6/2) = log₃ 3 Step 4: Combine the result with the third term. log₃ 3 − log₃ 27 = log₃ (3/27) Step 5: Simplify the fraction inside the logarithm. log₃ (1/9) Step 6: Express 1/9 as a power of 3. Since 9 = 3², then 1/9 = 3⁻². log₃ (3⁻²) Step 7: Use the definition of logarithm, log_b (b^x) = x. log₃ (3⁻²) = −2 The simplified expression is: −2

Simplify the logarithmic expression: (1)/(2)3 36 - 3 2 - 33 3

Answer

Step-by-step solution

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

Substitute these simplified terms back into the original expression.

Apply the quotient rule of logarithms, $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$ , from left to right.

Combine the result with the third term.

Simplify the fraction inside the logarithm.

Express $\frac{1}{9}$ as a power of 3.

Use the definition of logarithm, $\log_b (b^x) = x$ .

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