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log_((5x-5))5+log_((x-1))^(2)125 > 2

$\log _{(5 x-5)} 5+\log _{(x-1)}{ }^{2} 125>2$

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Product property of logarithms

Full solution

Q.

\log _{(5 x-5)} 5+\log _{(x-1)}{ }^{2} 125>2

Apply Base Change Formula: Simplify the logarithmic expressions using the base change formula. $\newline$ The base change formula states that $\log_b a = \frac{\log_c a}{\log_c b}$ . We can apply this to both logarithms in the inequality. $\newline$ For the first term, $\log_{(5x-5)}5$ , we can write it as $\frac{\log(5)}{\log(5x-5)}$ . $\newline$ For the second term, $\log_{(x-1)}^{2}125$ , we can write it as $2 \cdot \frac{\log(125)}{\log(x-1)}$ .
Simplify Constants: Recognize that $\log(5)$ and $\log(125)$ are constants and can be simplified. $\log(125)$ is equivalent to $\log(5^3)$ which simplifies to $3 \times \log(5)$ . So the inequality becomes $\frac{\log(5)}{\log(5x-5)} + 2 \times \left(3 \times \log(5)\right) / \log(x-1) > 2$ .
Combine Terms with Common Denominator: Combine the terms over a common denominator. $\newline$ To combine the terms, we need a common denominator which is $\log(5x-5) \cdot \log(x-1)$ . $\newline$ The inequality now becomes $\frac{\log(5) \cdot \log(x-1) + 6 \cdot \log(5) \cdot \log(5x-5)}{\log(5x-5) \cdot \log(x-1)} > 2$ .
Clear Fraction by Multiplication: Multiply both sides of the inequality by the positive denominator to clear the fraction. $\newline$ Multiplying both sides by $\log(5x-5) \cdot \log(x-1)$ gives us: $\newline$ $\log(5) \cdot \log(x-1) + 6 \cdot \log(5) \cdot \log(5x-5) > 2 \cdot \log(5x-5) \cdot \log(x-1)$ .
Distribute and Simplify: Distribute and simplify the inequality. $\newline$ Distribute $2$ on the right side of the inequality: $\newline$ $\log(5) \cdot \log(x-1) + 6 \cdot \log(5) \cdot \log(5x-5) > 2 \cdot \log(5x-5) \cdot \log(x-1)$ . $\newline$ This simplifies to: $\newline$ $\log(5) \cdot \log(x-1) + 6 \cdot \log(5) \cdot \log(5x-5) > 2 \cdot \log(5x-5) \cdot \log(x-1)$ .
Isolate Terms by Subtraction: Subtract $2 \times \log(5x-5) \times \log(x-1)$ from both sides to isolate terms. $\newline$ $\log(5) \times \log(x-1) + 6 \times \log(5) \times \log(5x-5) - 2 \times \log(5x-5) \times \log(x-1) > 0$ .
Factor Out Common Terms: Factor out the common terms and simplify the inequality. $\newline$ We can factor out $\log(5)$ from the left side: $\newline$ $\log(5) \times (\log(x-1) + 6 \times \log(5x-5) - 2 \times \log(5x-5) \times \log(x-1) / \log(5)) > 0$ .
Correct Mistake in Factoring: Recognize a mistake in the previous step and correct it. $\newline$ The previous step contains a mistake in the factoring process. We cannot factor out $\log(5)$ from terms that do not all contain it. We need to go back and correctly simplify the inequality.

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