If $log_{0.3}(x-1)

Q: If $log_{0.3}(x-1)<log_{0.09}(x-1)$, then x lies in the interval:

If $log_{0.3}(x-1)<log_{0.09}(x-1)$, then x lies in the interval:

Question

If $log_{0.3}(x-1)<log_{0.09}(x-1)$, then x lies in the interval:

Accepted Answer

To determine why Option 4 is the correct answer for the inequality $ \log_{0.3}(x-1) < \log_{0.09}(x-1) $, we need to analyze the behavior of logarithmic functions with different bases. ### Explanation: 1. **Understanding the Bases:** - The base of the first logarithm is $0.3$, which is less than 1. - The base of the second logarithm is $0.09$, which is also less than 1. - For logarithms with bases less than 1, the function is decreasing. This means that as the argument of the logarithm increases, the value of the logarithm decreases. 2. **Comparing the Bases:** - Since $0.09 = (0.3)^2$, the base $0.09$ is smaller than $0.3$. - For decreasing functions, a smaller base results in a steeper decrease. Therefore, $ \log_{0.09}(x-1) $ decreases faster than $ \log_{0.3}(x-1) $. 3. **Analyzing the Inequality:** - The inequality $ \log_{0.3}(x-1) < \log_{0.09}(x-1) $ implies that the value of $x-1$ must be such that the logarithm with the smaller base (0.09) is less than the logarithm with the larger base (0.3). - This occurs when $x-1$ is greater than 1, because for values greater than 1, the logarithm with the smaller base will have a smaller (more negative) value than the logarithm with the larger base. 4. **Solving for $x$:** - Since $x-1 > 1$, we have $x > 2$. 5. **Conclusion:** - Therefore, $x$ lies in the interval $(2, \infty)$, which corresponds to Option 4. ### Why Other Options Are Incorrect: - **Option 1: (-2, -1)** - In this interval, $x-1$ is negative, and logarithms of negative numbers are undefined in the real number system. Thus, this option is invalid. - **Option 2: (-1, 1)** - In this interval, $x-1$ ranges from -2 to 0, which again involves taking logarithms of negative numbers or zero, both of which are undefined. Hence, this option is incorrect. - **Option 3: (1, 2)** - In this interval, $x-1$ ranges from 0 to 1. For values between 0 and 1, the logarithms are defined, but the inequality $ \log_{0.3}(x-1) < \log_{0.09}(x-1) $ does not hold because both logarithms will be negative, and the one with the smaller base (0.09) will be more negative, contradicting the inequality. Thus, Option 4 is the only interval where the inequality holds true, making it the correct answer.