QuestionJuly 15, 2025

Find the largest value of x that satisfies: log_(6)(x^2)-log_(6)(x+2)=4 x=square

Find the largest value of x that satisfies: log_(6)(x^2)-log_(6)(x+2)=4 x=square
Find the largest value of x that satisfies: log_(6)(x^2)-log_(6)(x+2)=4 x=square

Solution
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Answer

x = 1296 Explanation 1. Apply Logarithm Property Use the property \log_{b}(A) - \log_{b}(B) = \log_{b}(\frac{A}{B}) to combine logs: \log_{6}(\frac{x^2}{x+2}) = 4. 2. Convert Logarithmic Equation to Exponential Form Convert to exponential form: \frac{x^2}{x+2} = 6^4. 3. Simplify and Solve Quadratic Equation Calculate 6^4 = 1296. Set up equation: x^2 = 1296(x + 2), leading to x^2 = 1296x + 2592. Rearrange: x^2 - 1296x - 2592 = 0. 4. Use Quadratic Formula Apply quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} with a=1, b=-1296, c=-2592: x = \frac{1296 \pm \sqrt{1296^2 + 4 \times 2592}}{2}. 5. Calculate Discriminant and Roots Calculate discriminant: 1296^2 + 4 \times 2592 = 1689216. Find roots: x = \frac{1296 \pm \sqrt{1689216}}{2}. 6. Determine Largest Root Calculate \sqrt{1689216} = 1296. Roots are x = \frac{1296 \pm 1296}{2}. Largest root: x = \frac{2592}{2} = 1296.

Explanation

1. Apply Logarithm Property<br /> Use the property $\log_{b}(A) - \log_{b}(B) = \log_{b}(\frac{A}{B})$ to combine logs: $\log_{6}(\frac{x^2}{x+2}) = 4$.<br />2. Convert Logarithmic Equation to Exponential Form<br /> Convert to exponential form: $\frac{x^2}{x+2} = 6^4$.<br />3. Simplify and Solve Quadratic Equation<br /> Calculate $6^4 = 1296$. Set up equation: $x^2 = 1296(x + 2)$, leading to $x^2 = 1296x + 2592$. Rearrange: $x^2 - 1296x - 2592 = 0$.<br />4. Use Quadratic Formula<br /> Apply quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-1296$, $c=-2592$: $x = \frac{1296 \pm \sqrt{1296^2 + 4 \times 2592}}{2}$.<br />5. Calculate Discriminant and Roots<br /> Calculate discriminant: $1296^2 + 4 \times 2592 = 1689216$. Find roots: $x = \frac{1296 \pm \sqrt{1689216}}{2}$.<br />6. Determine Largest Root<br /> Calculate $\sqrt{1689216} = 1296$. Roots are $x = \frac{1296 \pm 1296}{2}$. Largest root: $x = \frac{2592}{2} = 1296$.
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