Summary

• The exponential function $y=b^x$ is increasing if $b >1$ and decreasing if $0<b<1$
  • Its domain is $(- \infty, \infty)$
  • Its range is $(0, \infty)$
• The logarithmic function $y = \log_b(x)$ is the inverse of $y=b^x$
  • Its domain is $(0, \infty)$
  • Its range is $(- \infty, \infty)$
• The natural exponential function is $y = e^x$ and the natural logarithmic function is $y = \ln x$, or $y = \log _e x$
• Given an exponential function or logarithmic function in base $q$, we can make a change of base to convert this functionto any base $b > 0, b≠1$
  • We typically convert to base $e$
• The hyperbolic functions involvce combinations of the exponential functions $e^x$ and $e^{-x}$
  • As a result, the inverse hyperbolic functions involce the natural logarithm

Exponential Functions

<aside> 🔆 For a given positive number $a$, we can define an exponential function with base $a$ as $f(x) = a^x$. All exponential functions $a^x$ have their domain as the set of real numbers and their range and the set of positive real numbers

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Properties of Exponential Functions

• For larger values of the base $a$, the exponential function will have a higher growth rate

Euler’s Number

Logarithms

<aside> ☝ For each exponential function with base $a$, $a^x$, we define its inverse function called the logarithm base $a$, written as $\log _a (x)$ such that $\log _a (a^x) = a^{\log _a (x)} = x$

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• We also define the natural logarithm $\ln (x)$, which is a special notation for $log _e (x)$, thus, $\ln (e^x) = e^{\ln (x)} = x$
  • Derivative of $\ln(x) = \frac{1}{x}$
• All logarithmic functions $\log _a (x)$ have a domain of $x > 0$, the set of positive real numbers, and their range is given by the set of real numbers

Properties of Logarithms