 Growth: interest, births  Decay: isotopes, drug levels, temperature  Scales: Richter, pH, decibel levels Exponential and Logarithm Functions Functions.

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 Growth: interest, births  Decay: isotopes, drug levels, temperature  Scales: Richter, pH, decibel levels Exponential and Logarithm Functions Functions involved are called exponential and logarithmic.

Q A function when the base(a) is some positive number. Q The exponent is variable(x). Q The exponential function with base a is defined by: Exponential Functions

Example 1 x-2012 f(x)f(x)1/41/ x f (x) Domain: Range: Horizontal Asymptote:

x-2012 f(x)f(x)9311/31/ x f (x) Domain: Range: Horizontal Asymptote: Example 2

Special base, e  …….. Use a calculator to evaluate the following values of the natural exponential function (round to 5 decimal places): Natural Base, e

Q Exponential functions f (x) = a x are one-to-one functions. Q This means they each have an inverse function, a function that reverses what the original function did. Q We denote the inverse function with log a, the logarithmic function with base a, written as: Logarithmic Functions and we say “f of x is the logarithm of x base a”.

Exponential vs. logarithmic form Switch from logarithmic form to exponential form:

Switch from exponential form to logarithmic form: Evaluating logarithms

x y Domain restrictions (from first week): 1.No negatives under an even root 2.No division by zero 3.Only positives inside a logarithm 1 Domain : Range: Vertical Asymptote: Graph

1. log a a x = x (you must raise a to the power of x to get a x ) 2. a log a x = x (log a x is the power to which a must be raised to get x) Both are also the result of composing a function with its inverse. Properties of logarithms and exponentials

With calculator: Common Logarithm (Base 10) Without calculator:

With calculator: Without calculator: Natural Logarithm (Base e) To evaluate other bases on the calculator, use the following change of base formula: log a b

Q Isolate exponential function and apply logarithm function to both sides of the equation. Q Isolate the logarithm function and apply the base to both sides of the equation. Q Remember inverse properties and change of base: Solving equations

Example 1

Example 2

Example 3

(a) What is the initial number of bacteria? (b) What is the relative rate of growth? Express your answer as a percentage. (c) How many bacteria are in the culture after 5 hours? Please round the answer to the nearest integer. (d) When will the number of bacteria reach 10,000? Please round the answer to the nearest hundredth. Example 4

(a) How much remains after 60 days? (b) When will 10 grams remain? Please round the answer to the nearest day. (c) Find the half-life of polonium-210. Example 5 The mass m(t) remaining after t days from 40 gram sample of polonium-210 is given by: