Exponentials and logarithms

Section 4.8 Exponentials and logarithms

We now introduce exponentials and logarithms. Using what we learned about inverse functions and implicit differentiation, we calculate their derivatives. We also study logarithmic differentiation, which is a fancy application of implicit differentiation.

Objectives

You should be able to:

Sketch the graphs of exponential and logarithmic functions, and recall their main properties.
Derive the derivative of exponential and logarithmic functions from the definition of derivatives, inverse functions and implicit differentiation.
Differentiate functions involving exponential and logarithmic functions.
Differentiate functions involving products, quotients or powers by using logarithms (logarithmic differentiation).
Find the number \(e\) as a limit.

Subsection 4.8.1 Instructional video

Subsection 4.8.2 Key concepts

Concept 4.8.1. Exponential functions.

Exponential functions are functions of the form \(f(x) = a^x\) for some positive constant \(a\text{.}\) The domain of \(f(x)=a^x\text{,}\) for any \(a\text{,}\) is \(\mathbb{R}\text{,}\) while the range (for \(a \neq 1\)) is \((0,\infty)\) (as \(a^x\) is always positive for any real number \(x\)).

For \(a\) and \(b\) positive numbers and \(x\) and \(y\) real numbers, exponential functions satisfy:

\(a^{x+y} = a^x a^y\text{,}\)
\(a^{x-y} = \frac{a^x}{a^y}\text{,}\)
\((a^x)^y = a^{x y}\text{,}\)
\((ab)^x = a^x b^x\text{.}\)

Concept 4.8.2. The natural exponential function.

The base \(e \simeq 2.71828\) is such that the slope of the tangent line to \(y=e^x\) at \((0,1)\) is exactly one. The function \(f(x) = e^x\) is so cool that is has its own name: it is called the natural exponential function.

Concept 4.8.3. Logarithmic funcctions.

The logarithmic function with base \(a\), denoted by \(f(x) = \log_a(x)\text{,}\) is the inverse function of the exponential function \(a^x\text{.}\) That is,

\begin{equation*} x = a^y \qquad \text{if and only if} \qquad \log_a(x) = y. \end{equation*}

The domain of logarithmic functions (for \(a \neq 1\)) is \((0,\infty)\text{,}\) while the range is \(\mathbb{R}\) (as it is the inverse of the exponential function \(a^x\text{,}\) and so the domain and range are exchanged).

By definition of inverse functions, we have:

\begin{align*} \log_a(a^x) =\amp x \qquad \text{for }x \in \mathbb{R},\\ a^{\log_a(x)} =\amp x \qquad \text{for }x \in (0,\infty). \end{align*}

For \(x\) and \(y\) positive numbers, and \(r\) a real number, logarithmic functions satisfy:

\(\log_a(xy) = \log_a(x) + \log_a(y)\text{,}\)
\(\log_a\left( \frac{x}{y} \right) = \log_a(x) - \log_a(y)\text{,}\)
\(\displaystyle \log_a\left(x^r \right) = r \log_a(x).\)

Concept 4.8.4. The natural logarithm.

Just as the natural exponential function \(e^x\) is very cool, so is its inverse. The logarithm with base \(e\) is called the natural logarithm and is denoted by \(\ln(x) := \log_e(x)\text{.}\)

Concept 4.8.5. Change of base formula.

For any positive number \(a \neq 1\text{,}\)

\begin{equation*} \log_a(x) = \frac{\ln(x)}{\ln(a)}. \end{equation*}

Concept 4.8.6. Derivatives of exponential and logarithmic functions.

\begin{align*} \frac{d}{d x}( e^x) =\amp e^x,\\ \frac{d}{d x} (a^x) =\amp a^x \ln(a),\\ \frac{d}{d x}( \ln(x)) =\amp \frac{1}{x},\\ \frac{d}{d x} (\log_a(x) )=\amp \frac{1}{x \ln(a)}. \end{align*}

Concept 4.8.7. Logarithmic differentiation.

Suppose that you are given a function \(y=f(x)\text{.}\) The idea of logarithmic differentiation is to “take the logarithm and then differentiate”. More precisely, we first take the absolute value (as the argument of a logarithm must always be positive), and then take the natural logarithm on both sides of the relation to get

\begin{equation*} \ln|y| = \ln|f(x)| \text{.} \end{equation*}

Note that if the function \(f(x)\) is always positive, you can drop the absolute value. We then use implicit differentiation, i.e. we differentiate both sides with respect to \(x\text{,}\) considering that \(y\) is an arbitrary function of \(x\text{.}\) We get

\begin{equation*} \frac{y'}{y} = \frac{d}{dx} \ln |f(x)|. \end{equation*}

We then solve for \(y'\text{,}\) and substitute back \(y = f(x)\text{,}\) to get

\begin{equation*} y' = f(x) \frac{d}{dx} \ln |f(x)| . \end{equation*}

We can get \(y'\) by evaluating the remaining derivative on the right-hand-side, for a specific choice of \(f(x)\text{.}\)

This method is useful when it is easier to evaluate the derivative of \(\ln |f(x)|\) than of \(f(x)\text{:}\) for instance, for functions \(f(x)\) such that both the base and the exponent depend on \(x\) (for instance, consider the function \(f(x) = x^x\) with \(x>0\text{,}\) in which case \(\ln f(x)= \ln (x^x) = x \ln x\)), or for functions \(f(x)\) that are complicated products or quotients of functions (in which case logarithmic differentiation is faster than product and quotient rules).

MATH 144: Calculus for the Physical Sciences I