When working in the real numbers with log functions, remember to always follow the three rules listed above. We introduce the absolute value into the log to ensure that the antiderivative is defined everywhere the integral is. The absolute value is important because this is an indefinite integral, which means x might range through the entire real number line (There is a singularity at x=0, but log(0) is undefined too). This common integral can be found in many integral tables. Evaluate the indefinite integral в€« 1вЃ„x dx Therefore, to simplify this function, we must make sure the domain remains all real nonzero numbers. g(x,y) = log|x| + log|y| is defined for all real, nonzero values of x and y. It is important to notice that the above solution is not true in the reverse direction. Simplify the function g(x,y) = log|x| + log|y| Since the domain is different than that of log(xy), they cannot be equivalent functions.Д®xample 2. If we had not included the absolute values, log(x) + log(y) would only be defined when x and y are both positive (Quadrant I). To make sure that the values inside the log are positive, we introduce the absolute value. To expand, we must follow the above rules, with emphasis in this case on Rule 3. Therefore, the domain of our function is xy>0 which means that f(x,y) lies in Quadrants I and III on the x-y plane. We must always have a positive number inside the log function, and since x and y are being multiplied together, they must then always have the same sign. To prevent this from happening, we must follow these rules when working with real numbers, and to do so we often need to introduce the absolute value.Д®xample 1. If any of these conditions are not met, we risk introducing imaginary solutions or an infinite number of solutions. x cannot be less than or equal to zero.There are several restrictions on this definition when using real numbers (those not involving i = в€љ-1 ). (The double arrow is a bi-conditional, which means that one side is true if and only if the other side is true). When we define, exponents become thoroughly linked with trigonometric functions. But if is negative then we need to have so is a negative number. Suddenly being a negative number is not impossible. LetвЂ™s first review the definition of the logarithm function: This is Eulers Formula, one of the most famous mathematical formula in history. ThatвЂ™s because you canвЂ™t take the log of a negative number. Absolute values often turn up unexpectedly in problems involving logarithms.

Negative logs