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$\log(x+2)+\log(x)-1=\log\left(\frac{3}{2}\right)$

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Quotient property of logarithms

Full solution

Q.

\log(x+2)+\log(x)-1=\log\left(\frac{3}{2}\right)

Combine logarithms: Use the product rule for logarithms to combine $\log(x+2)$ and $\log(x)$ . $\log(x+2) + \log(x) = \log((x+2)\cdot x)$
Isolate logarithms: Subtract $1$ from both sides to isolate the logarithms. $\newline$ $\log((x+2)\cdot x) - 1 = \log\left(\frac{3}{2}\right)$
Convert to logarithmic form: Convert the subtraction of $1$ into the logarithmic form. $\log((x+2)\cdot x) - \log(10) = \log\left(\frac{3}{2}\right)$
Combine using quotient rule: Use the quotient rule for logarithms to combine the left side. $\log\left(\frac{(x+2)x}{10}\right) = \log\left(\frac{3}{2}\right)$
Set arguments equal: Since the logarithms are equal, their arguments must be equal. $\frac{(x+2)x}{10} = \frac{3}{2}$
Cross-multiply: Cross-multiply to solve for $x$ . $2(x+2)\cdot x = 3\cdot 10$
Distribute and simplify: Distribute and simplify the equation. $2x^2 + 4x = 30$
Move terms to one side: Move all terms to one side to form a quadratic equation. $2x^2 + 4x - 30 = 0$
Divide by $2$ : Divide the entire equation by $2$ to simplify. $\newline$ $x^2 + 2x - 15 = 0$
Factor quadratic equation: Factor the quadratic equation. $\newline$ $(x+5)(x-3) = 0$
Solve for x: Set each factor equal to zero and solve for x. $\newline$ $x+5 = 0$ or $x-3 = 0$
Final solutions: Solve each equation for $x$ . $x = -5$ or $x = 3$

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