Simplify: 2log₃6 + log₃12 - log₃16

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Apply the power rule of logarithms, a log_b x = log_b x^a, to the first term. 2 log₃ 6 + log₃ 12 − log₃ 16 = log₃ 6² + log₃ 12 − log₃ 16; = log₃ 36 + log₃ 12 − log₃ 16; Step 2: Apply the product rule of logarithms, log_b x + log_b y = log_b (x · y), and the quotient rule, log_b x − log_b y = log_b (x/y). = log₃ ((36 × 12)/(16)) Step 3: Simplify the expression inside the logarithm. = log₃ (432/16); = log₃ 27 Step 4: Evaluate the logarithm. We need to find the power to which 3 must be raised to get 27. Since 3³ = 27, we have: = log₃ 3³; = 3 log₃ 3; Since log₃ 3 = 1: = 3 × 1; = 3 The final answer is 3.

Simplify: 2log₃6 + log₃12 - log₃16

Answer

Step-by-step solution

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

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

Simplify the expression inside the logarithm.

Evaluate the logarithm.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Simplify: 2log₃6 + log₃12 - log₃16

Answer

Step-by-step solution

Apply the power rule of logarithms, alog⁡bx=log⁡bxaa \log_b x = \log_b x^aalogb​x=logb​xa, to the first term.

Apply the product rule of logarithms, log⁡bx+log⁡by=log⁡b(x⋅y)\log_b x + \log_b y = \log_b (x \cdot y)logb​x+logb​y=logb​(x⋅y), and the quotient rule, log⁡bx−log⁡by=log⁡b(xy)\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)logb​x−logb​y=logb​(yx​).

Simplify the expression inside the logarithm.

Evaluate the logarithm.

Try one yourself

Still stuck on this one?

More Mathematics Questions

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

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