Key Concepts

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

Glossary

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