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Exponential and Logarithmic Functions. Exponential Functions. Definition: If a is some number greater than 0, and a = 1, then the exponential function with base a is:. Examples:. Properties of Exponents. If a and b are positive numbers:. Graphs of Exponential Functions.

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## Exponential and Logarithmic Functions

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**Exponential and Logarithmic Functions**Session 6 : 9/22**Exponential Functions**• Definition: If a is some number greater than 0, and a = 1, then the exponential function with base a is: Examples: Session 6 : 9/22**Properties of Exponents**• If a and b are positive numbers: Session 6 : 9/22**Graphs of Exponential Functions**• Point plotting or graphing tool • If base is raised to positive x, function is an increasing exponential. If base is raised to negative x, function is a decreasing exponential • If a>1 and to a (+)x : Increasing Exponential • If a<1 and to a (+)x : Decreasing Exponential • If a>1 and to a (-)x: Decreasing Exponential • If a<1 and to a (-)x: Increasing Exponential Session 6 : 9/22**Sketching an Exponential**• Find horizontal asymptote and plot several points • How do we find horizontal asymptote? • Take the limit as x approaches infinite (for decreasing exponentials) or negative infinite (for increasing exponentials) Session 6 : 9/22**INCREASING EXPONENTIAL**8 x - Asymptote for increasing exponential function 8 x + Asymptote for decreasing exponential function Session 6 : 9/22**Natural Exponential Functions**• In calculus, the most convenient (or natural) base for an exponential function is the irrational number e (will become more obvious once we start trying to differentiate/integrate…) • e ≈ 2.718 • Simplest Natural Exponential: Session 6 : 9/22**Graph of the Natural Exponential Function**Session 6 : 9/22**Exponential Growth**• Exponential functions (particularly natural exponentials) are commonly used to model growth of a quantity or a population • What growth is unrestricted, can be described by a form of the standard exponential function (probably will have multiplying constants, slight changes…): • When growth is restricted, growth may be best described by the logistic growth function: Where a, b, and k are constants defined for a given population under specified conditions. Session 6 : 9/22**Comparing Exponential v. Logistic Growth Function**y Exponential Logistic Growth Function x Session 6 : 9/22**Derivatives of The Natural Exponential Function**• From now on, ‘Exponential Function’ will imply an function with base e • Previously, we said that e is the most convenient base to use in calculus. Why? • Very simple derivative! Chain rule, where u is a function of x Session 6 : 9/22**What does this mean graphically?**For the function the slope at any point x is given by the derivative slope = e2 slope =e1 slope =e0=1 1 2 Session 6 : 9/22**Examples:**Session 6 : 9/22**Logarithmic Functions**• Review of ‘log’ If no base specified, log10 Session 6 : 9/22**The Natural Log**• Natural Log=loge=ln • Definition of the natural log: The natural logarithmic function, denoted by ln(x), is defined as: Why? Session 6 : 9/22**Important Properties of Logarithmic Functions**Natural log is inverse of exponential Exponential is inverse of natural log Session 6 : 9/22**Examples:**• Solve the following logarithmic functions for x • Simplify the following: Session 6 : 9/22**Examples: Solving Exponential and Logarithmic Equations**Session 6 : 9/22**Example: Doubling Time**• For an account with initial balance P, the function for the account balance (A) after t years (with annual interest rate r compounded continuously) is given by: Find an expression for the time at which the account balance has doubled. Session 6 : 9/22**Derivative of logarithmic functions:**Where u is a function of x Session 6 : 9/22**Examples**Session 6 : 9/22**Exponential Growth and Decay**• Law of exponential growth and decay: If y is a positive quantity whose rate of change with respect to time is proportional to the quantity present at any time t, then y is described by: Where C is the initial value k is the constant of proportionality (often rate constant) If k > 0: Exponential Growth If k < 0: Exponential Decay Session 6 : 9/22**Example:**• Modeling population growth: • A researcher is trying to develop an equation to describe bacterial growth, and knows that it will follow the fundamental equation for exponential growth. The following data is available: • At t=2 hours, there are 1x106 cells • At t=8 hours, there are 5x108 cells • Write an equation for the exponential growth of bacterial cells by the following steps: • 1. Find k • 2. Find C using the solution for k • 3. Write the full model by plugging in C and k values. • Find the time at which the population is double that of the initial population. Session 6 : 9/22

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