5 log₉ y − 2 log₉ (x+4) = 2 log₉ y + log₉ x

Question

ScanSolve AI Tutor · Accepted Answer

To make y the subject of the formula, we will use the properties of logarithms. The given equation is: 5 log_a y − 2 log_a (x+4) = 2 log_a y + log_a x; Step 1: Apply the power rule of logarithms, n log_b M = log_b Mⁿ, to all terms. log_a y⁵ − log_a (x+4)² = log_a y² + log_a x Step 2: Group terms involving log_a y on one side and terms involving log_a x and log_a (x+4) on the other side. log_a y⁵ − log_a y² = log_a x + log_a (x+4)²; Step 3: Apply the quotient rule, log_b M − log_b N = log_b (M/N), to the left side and the product rule, log_b M + log_b N = log_b (MN), to the right side. log_a ((y⁵)/(y²)) = log_a (x(x+4)²); log_a y⁵⁻² = log_a (x(x+4)²); log_a y³ = log_a (x(x+4)²) Step 4: Since the logarithms have the same base on both sides, we can equate their arguments. y³ = x(x+4)² Step 5: Solve for y by taking the cube root of both sides. y = √[3]x(x+4)²; The final answer is y = √[3]x(x+4)².

5 log_9 y - 2 log_9 (x+4) = 2 log_9 y + log_9 x

Answer

Step-by-step solution

Apply the power rule of logarithms, $n \log_b M = \log_b M^n$ , to all terms.

Group terms involving $\log_a y$ on one side and terms involving $\log_a x$ and $\log_a (x+4)$ on the other side.

Apply the quotient rule, $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$ , to the left side and the product rule, $\log_b M + \log_b N = \log_b (MN)$ , to the right side.

Since the logarithms have the same base on both sides, we can equate their arguments.

Solve for $y$ by taking the cube root of both sides.

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