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Find the domain of the following logarithmic function
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Q. Find the domain of the following logarithmic function
- Understand Logarithmic Function Domain: Understand the domain of a logarithmic function. The domain of a logarithmic function is all the values of for which the argument of the logarithm, , is greater than zero. This is because the logarithm of a non-positive number is undefined.
- Set Up Inequality for Domain: Set up the inequality to find the domain.For , we need to find the values of for which the argument of the logarithm, , is greater than zero.So, we set up the inequality:\frac{2x-4}{x+3}-5 > 0
- Simplify the Inequality: Simplify the inequality.To solve the inequality (2x-4)/(x+3)-5 > 0, we first combine the terms over a common denominator:(2x-4-5(x+3))/(x+3) > 0Simplifying the numerator gives us:(2x-4-5x-15)/(x+3) > 0Which simplifies to:(-3x-19)/(x+3) > 0
- Find Critical Points: Find the critical points of the inequality.The critical points are the values of that make the numerator zero or the denominator undefined. For the numerator , we have . For the denominator , we have . These are the points where the sign of the inequality can change.
- Test Intervals for Inequality: Test intervals around the critical points.We have two critical points, and . We need to test values in the intervals , , and to see where the inequality \frac{-3x-19}{x+3} > 0 holds true.
- Test Interval : Test the interval . Choose a test point (which is less than . Plugging this into the inequality gives: \frac{-3(-7)-19}{-7+3} > 0 \frac{21-19}{-4} > 0 \frac{2}{-4} > 0 This is false, so the interval is not part of the domain.
- Test Interval : Test the interval . Choose a test point (which is between and ). Plugging this into the inequality gives: (-3(-4)-19)/(-4+3) > 0 (12-19)/(-1) > 0 (-7)/(-1) > 0 7 > 0 This is true, so the interval is part of the domain.
- Test Interval Test the interval . Choose a test point (which is greater than ). Plugging this into the inequality gives: (-3(0)-19)/(0+3) > 0 (-19)/3 > 0 This is false, so the interval is not part of the domain.
- Combine True Intervals: Combine the intervals where the inequality is true.From our testing, we found that the interval is the only part of the domain. However, we must exclude the value because it makes the denominator zero, which is undefined.
