Key Concepts

• Exponential functions: [latex]f(x)=a \cdot b^{x}[/latex] where [latex]b>0, b \neq 1[/latex]. The value of the variable [latex]x[/latex] can be any real number.
• Continuous growth/decay is modeled by [latex]A(t)=a \cdot e^{rt}[/latex], for all real numbers [latex]r,t,[/latex] and all positive numbers [latex]a[/latex];
  • [latex]a[/latex] is the initial value
  • [latex]r[/latex] is the continuous growth [latex](r>0)[/latex] or decay [latex](r<0)[/latex] rate per unit time
  • and [latex]t[/latex] is the elapsed time.
• One-to-one property of exponential functions: [latex]b^{x}=b^{y}[/latex] if and only if [latex]x=y[/latex], where [latex]b>0, b \neq 1[/latex].
• Logarithmic function with base [latex]b[/latex]: For [latex]b>0, b \neq 1, y = \mathrm{log}_{b} \ x[/latex] if and only if [latex]b^{y}=x[/latex], where [latex]x>0[/latex].
  • [latex]y=\mathrm{log}_b \ x[/latex] os the logarithmic form
  • [latex]b^{y} = x[/latex] is the exponential form
• Exponential equations may be solve by
  • The one-to-one property: [latex]b^{S}=b^{T}[/latex] if and only if [latex]S=T[/latex], or
  • By isolating the exponential expression and writing in logarithmic form. Then solve for the variable.

Glossary

• common logarithms: have an implied base [latex]b=10[/latex]: [latex]\mathrm{log}(x) = \mathrm{log}_{10} (x)[/latex
• continuous random variables: variables that can take on any value within a range of values
• [latex]\boldsymbol{e}[/latex]: the irrational number which is the limiting value of [latex](1+ \frac{1}{n})^{n}[/latex] as [latex]n[/latex] increases without bound, [latex]e \approx 2.718282[/latex]
• exponential growth: quantity grows by a rate proportional to the current amount
• natural logarithms: have base [latex]e[/latex]: [latex]\mathrm{ln} (x) = \mathrm{log}_{e} (x)[/latex]