Exponential Functions

The exponential function \(f\) with base \(a\) (where \(a>0, a \neq 1\) ) is defined for all real numbers \(x\) by

\[ f(x)=a^{x} \]

The domain of \(f\) is \(\mathbb{R}\), and the range of \(f\) is \((0, \infty)\) The graph of \(f\) has one of the following shapes, depending on the value of \(a\) :

\(f(x)=a^{x}\) for \(a>1\)

\(f(x)=a^{x}\) for \(0<a<1\)

The Natural Exponential Function

The natural exponential function is the exponential function with base \(e\) :

\[ f(x)=e^{x} \]

The number \(e\) is defined to be the number that the expression \((1+1 / n)^{n}\) approaches as \(n \rightarrow \infty\). An approximate value for the irrational number \(e\) is

\[ e \approx 2.7182818284590 \ldots \]

Compound Interest

If a principal \(P\) is invested in an account paying an annual interest rate \(r\), compounded \(n\) times a year, then after \(t\) years the amount \(A(t)\) in the account is

\[ A(t)=P\left(1+\frac{r}{n}\right)^{n t} \]

If the interest is compounded continuously, then the amount is

\[ A(t)=P e^{r t} \]

Logarithmic Functions

The logarithmic function \(\log _{a}\) with base \(a\) (where \(a>0, a \neq 1\) ) is defined for \(x>0\) by

\[ \log _{a} x=y \quad \Leftrightarrow \quad a^{y}=x \]

So \(\log _{a} x\) is the exponent to which the base \(a\) must be raised to give \(y\).

The domain of \(\log _{a}\) is \((0, \infty)\), and the range is \(\mathbb{R}\). For \(a>1\), the graph of the function \(\log _{a}\) has the following shape:

\[ y=\log _{a} x \text { for } a>1 \]

Common and Natural Logarithms

The logarithm function with base 10 is called the common logarithm and is denoted log. So

\[ \log x=\log _{10} x \]

The logarithm function with base \(e\) is called the natural logarithm and is denoted \(\mathbf{l n}\). So

\[ \ln x=\log _{e} x \]

Properties of Logarithms

  1. \(\log _{a} 1=0\)
  2. \(\log _{a} a=1\)
  3. \(\log _{a} a^{x}=x\)
  4. \(a^{\log _{a} x}=x\)

Laws of Logarithms

Let \(a\) be a logarithm base \((a>0, a \neq 1)\), and let \(A, B\), and \(C\) be any real numbers or algebraic expressions that represent real numbers, with \(A>0\) and \(B>0\). Then:

  1. \(\log _{a}(A B)=\log _{a} A+\log _{a} B\)
  2. \(\log _{a}(A / B)=\log _{a} A-\log _{a} B\)
  3. \(\log _{a}\left(A^{C}\right)=C \log _{a} A\)

Change of Base Formula

\[ \log _{b} x=\frac{\log _{a} x}{\log _{a} b} \]

Guidelines for Solving Exponential Equations

  1. Isolate the exponential term on one side of the equation.
  2. Take the logarithm of each side, and use the Laws of Logarithms to "bring down the exponent."
  3. Solve for the variable.

Guidelines for Solving Logarithmic Equations

  1. Isolate the logarithmic term(s) on one side of the equation, and use the Laws of Logarithms to combine logarithmic terms if necessary.
  2. Rewrite the equation in exponential form.
  3. Solve for the variable.

Exponential Growth Model

A population experiences exponential growth if it can be modeled by the exponential function

\[ n(t)=n_{0} e^{r t} \]

where \(n(t)\) is the population at time \(t, n_{0}\) is the initial population (at time \(t=0\) ), and \(r\) is the relative growth rate (expressed as a proportion of the population).

Logistic Growth Model

A population experiences logistic growth if it can be modeled by a function of the form

\[ n(t)=\frac{M}{1+A e^{-r t}} \]

where \(n(t)\) is the population at time \(t, r\) is the initial relative growth rate, \(M\) is the carrying capacity of the environment, and \(A=\left(M-n_{0}\right) / n_{0}\), where \(n_{0}\) is the initial population.

Radioactive Decay Model

If a radioactive substance with half-life \(h\) has initial mass \(m_{0}\), then at time \(t\) the mass \(m(t)\) of the substance that remains is modeled by the exponential function

\[ m(t)=m_{0} e^{-r t} \]

where \(r=\frac{\ln 2}{h}\).

Newton's Law of Cooling

If an object has an initial temperature that is \(D_{0}\) degrees warmer than the surrounding temperature \(T_{s}\), then at time \(t\) the temperature \(T(t)\) of the object is modeled by the function

\[ T(t)=T_{s}+D_{0} e^{-k t} \]

where the constant \(k>0\) depends on the size and type of the object.

Logarithmic Scales

The \(\mathbf{p H}\) scale measures the acidity of a solution:

\[ \mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] \]

The Richter scale measures the intensity of earthquakes:

\[ M=\log \frac{I}{S} \]

The decibel scale measures the intensity of sound:

\[ B=10 \log \frac{I}{I_{0}} \]