Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the domain, any intercepts, and any asymptotes of the function. \(y=\ln x^{4}\)

Step-by-Step Solution

Answer

The domain of the function \(y=\ln x^{4}\) is \(-\infty < x < 0\) or \(0 < x < \infty\), the function's intercepts occur at \(x=1\) and \(x = -1\), and \(x = 0\) is a vertical asymptote of the function.

Step 1: Determine the domain of the function

The domain of the function is the set of all real numbers \(x\) for which the function is defined. \(y=\ln x^{4}\) is defined for any \(x^{4}> 0 \), which implies \(x\neq 0\). So, the domain of the function is \(-\infty < x < 0\) or \(0 < x < \infty\).

Step 2: Find any intercepts of the function

An intercept of the function is a point of intersection \(x\) where \(y = 0\). Solving the equation \(\ln x^{4}=0\), the logarithm base e is zero when its argument is 1. Thus, the function's intercepts occur at \(x = 1\) and \(x = -1\)

Step 3: Identify any asymptotes of the function

A vertical asymptote is a value of \(x\) that the function approaches but never reaches. For the function \(\ln x^{4}\), the function approaches negative infinity as \(x\) approaches 0 from the right and the left. Thus, \(x = 0\) is a vertical asymptote of the function.

Key Concepts

Domain of a Function

Understanding the domain of a function is essential for graphing and analyzing mathematical expressions. The domain includes all the values for which the function is mathematically defined. In the case of the logarithmic function y = \(\ln x^4\), we need to consider values of x that make \(\ln x^4\) exist. Since logarithms are undefined for non-positive numbers, the value inside the logarithm, \(x^4\), must be greater than zero. As a result, x cannot be zero. Therefore, the domain is all real numbers except zero, or in interval notation: \((-\infty, 0) \cup (0, +\infty)\).

Remember, logarithms only take positive arguments, which sets the basic domain for all logarithmic functions. However, since we are dealing with \(x^4\), every real number except zero will give a positive result when raised to the fourth power, thus further defining the domain of this specific function.

Intercepts of a Function

Intercepts are points where the graph of a function crosses the axes. In particular, the x-intercepts occur where the function equals zero. For our given function y = \(\ln x^4\), finding the x-intercept means solving \(\ln x^4 = 0\). Recall that the natural logarithm of 1 is 0, which implies \(x^4 = 1\). Therefore, the intercepts occur at x = 1 and x = -1, since both fourth powers equal 1.

Y-intercepts, on the other hand, are found by setting x to zero. However, in this function, x cannot be zero as discussed in the domain section, hence there is no y-intercept. It's useful to note that logarithmic functions often don't have a y-intercept because they are undefined at x = 0.

Asymptotes

An asymptote is a line that the graph of a function approaches but never actually reaches. Logarithmic functions typically have vertical asymptotes, which occur where the function grows without bound.

In the problem y = \(\ln x^4\), as x gets closer and closer to zero from either side, the function goes to negative infinity. Therefore, the vertical asymptote for this function is the y axis, or x = 0. It's pertinent to understand that a logarithmic function will never touch or cross its vertical asymptote, symbolizing that the argument of the logarithm never reaches zero.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. This function is the inverse of the exponential function e^x and, thus, has intrinsic connections to growth and decay processes in mathematics.

\(\ln x\) only takes positive arguments, reflecting the idea that we can only take the logarithm of positive numbers. This specificity impacts the features of the graph, including its domain, intercepts, and asymptotes. In calculus, the natural logarithm is particularly important due to its derivative and integral properties, which play key roles in solving a wide range of problems.