Exponential, logarithmic and inverse functions

Polynomials and rational functions describe many engineering quantities, but some of the most important processes in nature and technology follow a different pattern: a quantity changes at a rate proportional to its current size. A charged capacitor discharges this way, a radioactive sample decays this way, a hot casting cools roughly this way, and money grows this way under compound interest. These processes are described by exponential functions, and undoing them, finding how long it takes to reach a given value, requires their inverses, the logarithms. This lesson develops the idea of an inverse function, then studies exponential and logarithmic functions, the inverse trigonometric functions that turn ratios back into angles, and the limits of these functions that you will need when you differentiate them in Lessons 4 and 5. It also shows how to recognize exponential behavior in measured data.

One-to-one functions and inverses

A function \(f\) takes each input \(x\) to one output \(f(x)\). An inverse function \(f^{-1}\) runs the process backward, taking each output back to the input that produced it:

An inverse exists only if no two inputs give the same output, that is, if \(f\) is one-to-one. Graphically, a function is one-to-one if every horizontal line crosses its graph at most once (the horizontal line test). The function \(f(x) = x^3\) passes: each output comes from exactly one input. The function \(f(x) = x^2\) fails, because \(f(2) = f(-2) = 4\), so the question "which input gave 4?" has two answers. We can still define an inverse by restricting the domain: on \(x \ge 0\), \(x^2\) is one-to-one and its inverse is \(\sqrt{x}\).

To find an inverse algebraically, write \(y = f(x)\), solve for \(x\) in terms of \(y\), then rename the variables if you want the inverse as a function of \(x\). The composition of a function and its inverse gives back the original input: \(f^{-1}(f(x)) = x\) for every \(x\) in the domain of \(f\), and \(f(f^{-1}(y)) = y\) for every \(y\) in its range. The domain of \(f^{-1}\) is the range of \(f\), and vice versa.

Because the inverse swaps the roles of input and output, the graph of \(f^{-1}\) is the mirror image of the graph of \(f\) in the line \(y = x\).

Exponential functions

An exponential function has the variable in the exponent:

For \(a > 1\) it increases, slowly at first and then ever faster; for \(0 < a < 1\) it decreases toward zero. Every exponential function passes through \((0, 1)\), is always positive, and has the horizontal asymptote \(y = 0\) on one side. The familiar laws of exponents apply: \(a^{x+y} = a^x a^y\), \(a^{x-y} = a^x / a^y\), \((a^x)^y = a^{xy}\) and \(a^{-x} = 1/a^x\).

The key property of exponential growth is that equal steps in \(x\) multiply the output by equal factors. Increasing \(x\) by 1 multiplies \(2^x\) by 2, whatever the starting point. This is why exponential quantities have a fixed doubling time or half-life.

The number e

One base is special. Consider an account paying 100 percent interest per year, compounded \(n\) times a year. After one year, one dollar becomes \((1 + 1/n)^n\) dollars. Compounding more often gives more, but the amount approaches a limit:

The function \(e^x\), often written \(\exp(x)\), is called the natural exponential function. Lesson 4 will show why it is natural: it is the only exponential function whose slope at every point equals its own value, which makes it the function that describes any process whose rate of change is proportional to its size. Every exponential can be written with base \(e\), since \(a^x = e^{x \ln a}\).

Exponential models

A quantity that grows or decays at a rate proportional to its size follows:

where \(Q_0\) is the initial value and \(k\) the growth rate (positive for growth, negative for decay). Decay problems are often written with a time constant \(\tau = -1/k\), so that \(Q = Q_0 e^{-t/\tau}\): after one time constant, the quantity has fallen to \(1/e \approx 36.8\) percent of its starting value.

Logarithms

Because \(a^x\) is one-to-one, it has an inverse, the logarithm to base \(a\):

In words, \(\log_a y\) is the exponent to which \(a\) must be raised to give \(y\). Logarithms are defined only for positive \(y\), because \(a^x\) is always positive. The logarithm to base \(e\) is the natural logarithm, written \(\ln\), and the logarithm to base 10 is the common logarithm, often written \(\log\). As inverse functions, \(\ln(e^x) = x\) for all \(x\) and \(e^{\ln y} = y\) for \(y > 0\). The graph of \(\ln x\) is the reflection of \(e^x\) in the line \(y = x\): it passes through \((1, 0)\), increases without bound but very slowly, and plunges toward \(-\infty\) as \(x \to 0^+\).

The laws of logarithms follow from the laws of exponents. For positive \(x\) and \(y\):

A logarithm to any base can be computed from natural logarithms with the change-of-base formula, \(\log_a x = \ln x / \ln a\).

Solving exponential equations

Logarithms bring exponents down to where they can be solved for. To solve \(Q_0 e^{kt} = Q\), divide by \(Q_0\) and take the natural logarithm of both sides: \(kt = \ln(Q/Q_0)\), so \(t = \ln(Q/Q_0)/k\). Two useful special cases are the doubling time of growth, \(t_2 = \ln 2 / k \approx 0.693/k\), and the half-life of decay, \(t_{1/2} = \ln 2 / |k|\).

Logarithmic scales

Quantities that span many orders of magnitude are often measured on logarithmic scales. Sound pressure level in decibels is \(L = 20 \log_{10}(p/p_0)\), where \(p_0 = 20\) µPa, so every factor of 10 in sound pressure adds 20 dB (and every factor of 10 in sound power adds 10 dB). The pH of a solution is \(-\log_{10}[\text{H}^+]\). Earthquake magnitudes, signal gains in electronics, and star brightnesses are all logarithmic. On such scales, equal steps represent equal ratios, not equal differences.

Inverse trigonometric functions

The sine function is not one-to-one, because it repeats every \(2\pi\). To define an inverse, we restrict its domain to \([-\pi/2, \pi/2]\), where it increases through every value from −1 to 1 exactly once. The inverse sine, \(\arcsin x\) or \(\sin^{-1} x\), is the angle in \([-\pi/2, \pi/2]\) whose sine is \(x\). Similarly, \(\arccos x\) is the angle in \([0, \pi]\) whose cosine is \(x\), and \(\arctan x\) is the angle in \((-\pi/2, \pi/2)\) whose tangent is \(x\); \(\arctan x\) is defined for all real \(x\) and approaches \(\pm\pi/2\) as \(x \to \pm\infty\).

Note that \(\sin^{-1} x\) means the inverse function, not \(1/\sin x\), which is \(\csc x\). Calculators return values only in the restricted ranges, so the engineer must decide which angle a problem actually needs. To find the direction of a vector \((x, y)\) in any quadrant, programming languages provide a two-argument function, usually called atan2, that uses the signs of both components.

Limits of exponential and logarithmic functions

The limits of these functions at infinity, and their comparison with powers, appear constantly in engineering analysis:

• \(\lim_{x \to \infty} e^x = \infty\) and \(\lim_{x \to -\infty} e^x = 0\), so \(e^{-t/\tau} \to 0\) as \(t \to \infty\): every exponential decay dies out.
• \(\lim_{x \to \infty} \ln x = \infty\) and \(\lim_{x \to 0^+} \ln x = -\infty\).
• Exponentials beat powers: for any power \(n\), \(\lim_{x \to \infty} x^n / e^x = 0\). However large \(n\) is, \(e^x\) eventually grows faster.
• Powers beat logarithms: for any \(p > 0\), \(\lim_{x \to \infty} \ln x / x^p = 0\).

These comparisons can be checked numerically now and will be proved with L'Hôpital's rule in Lesson 7. They explain, for instance, why an algorithm whose running time grows exponentially with the size of its input soon becomes impractical, while one that grows like a power can scale, and one that grows like a logarithm barely notices larger inputs.

Recognizing exponential behavior in data

If \(y = A e^{kt}\), then taking logarithms gives \(\ln y = \ln A + kt\): a straight line when \(\ln y\) is plotted against \(t\). Plotting measured data on a semi-logarithmic graph (logarithmic vertical axis, linear horizontal axis) is therefore a quick test for exponential behavior, and the slope gives the rate \(k\). Similarly, a power law \(y = A x^n\) becomes \(\ln y = \ln A + n \ln x\), a straight line on a log-log graph with slope \(n\). These transformations are widely used in engineering to fit models to test data.