﻿ How to find the domain of a function with logarithm and square root?
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# How to find the domain of a function with square root and logarithm?

The numerator of the function is a square root expression. Because any positive number and zero can have a square root, so the number inside the square root must be large and equal to zero. That is, 5 - |x - 3| >= 0. This is our first inequality. Look the denominator lg (8 - 2x), one of the properties of logarithm is: negative number and zero do not have logarithm, so we get 8 - 2x > 0. This is our second inequality. If the denominator of a fraction is zero, then the fraction will be undefined. So, the denominator of a fraction cannot be zero. Another property of the logarithm is: the logarithm of one is zero. That is, log 1 = 0. So, 8 - 2x cannot be one. That is, 8 - 2x not equal to 1. This is our third condition. The domain of the function must satisfy the three conditions. Solving the first inequality involved in absolute value inequality. 5 - |x - 3| >= 0. Move the -|x - 3| to the right side of the inequality. We get |x - 3| <= 5. Apply the property of the absolute value inequality, -5 <= x - 3 <= 5. Add 3 to each side of the inequalities, -2 <= x <= 8. So, x must be in the range of [-2, 8]. In second inequality, 8 - 2x < 0. Move the -2x to the right side of the inequality, we get 2x < 8. Note: 8 is equal to 23. So, 2x < 23. Then x < 3. Combine the first inequality and the second inequality, we get that x must be in the range of [-2, 3]. Look the third condition: 8 - 2x not equal to 1. Move 1 to the left side of the not equal sign and move -2x to the right side of the not equal sign. We get that 8 - 1 not equal to 2x. That is, 7 not equal to 2x. Apply logarithm, then log 7 not equal to x log2, then x not equal to log7 ÷ log2. So, the domain of y is its independent variable x in the range of [-2, 3] and x not equal to log7 ÷ log2.