# Exponential and Logarithmic Functions

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A series of free, online Intermediate Algebra Lessons or Algebra II lessons.

Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

- exponential functions and their graphs
- how to solve exponential functions with the same base
- about logarithms
- how to solve basic logarithmic equations
- function notation with logs and exponentials
- how to graph of logarithmic functions

### Exponential Functions and their Graphs

There are certain functions, such as exponential functions, that have many applications to the real world and have useful inverse functions. Graphing exponential functions is used frequently, we often hear of situations that have exponential growth or exponential decay. The inverses of exponential functions are logarithmic functions. The graphs of exponential functions are used to analyze and interpret data.

A discussion on exponential functions and their graphs.

Graph Exponential Functions

### Solving Exponential Equations with the 'Same' Base

In the real world we often hear terms like exponential growth or exponential decay, when discussing solving exponential equations such as those used in compounding interest problems. In order to understand solving exponential equations, students should understand the significance of exponential functions and logarithmic functions.

Solving Exponential Equations

This video explains how to solve exponential equations when you can write each side of the equation with a common base. These equations to not require the use of logarithms.

Ex 4: Solve Exponential Equations Using Like Bases - No Logarithms

This video provides two examples of how to solve two exponential equations using like bases and the properties of exponents. Logarithms are not used.

Ex 5: Solve Exponential Equations Using Like Bases - No Logarithms

This video provides two examples of how to solve two exponential equations using like bases and the properties of exponents. Logarithms are not used.

Ex 6: Solve Exponential Equations Using Like Bases - No Logarithms

This video provides two examples of how to solve two exponential equations using like bases and the properties of exponents. Logarithms are not used.

### Introduction to Logarithms

The importance of having inverse functions leads us to the introduction to logarithms as the inverses of exponential functions. Solving exponential equations often involves using the ideas presented in the introduction to logarithms to simplify or access the variable for manipulation. To take the log of an exponential function helps us to use the exponentiated term.

This video defines a logarithms and provides examples of how to convert between exponential equations and logarithmic equations.

An introduction to logarithms

### Solving Basic Logarithmic Equations

In order to understand solving logarithmic equations, students must understand the basics of logarithms, and how to use exponentiation to access the terms inside the logarithm. Some more complicated instances of solving basic logarithmic equations require knowledge of the product, quotient and power rules of logarithms in order to simplify complex terms.

This video provides 4 examples of how to solve basic logarithmic equations by writing them as exponential equations and then solving for x.

This video explains how to solve a basic logarithmic equation by writing the equation as an exponential equation and solving for x. The exponential equation requires the use of radicals.

This video explains how to solve a basic logarithmic equation by writing the equation as an exponential equation and solving for x. The exponential equation ends up being a linear equation in one example and a quadratic equation in the other.

### Function Notation with Logs and Exponentials

Function notation is used frequently in science to express functions that contain logs and exponents. We learn to use function notation with logs and exponentials in order to solve problems such as computing compounding interest. We can solve these problems written in function notation with logs and exponentials using techniques from solving exponential and log equations.

How to use function notation to solve log equations.

### Graph of Logarithmic Functions

In science classes we will often find ourselves graphing logarithmic functions to describe situations such as motion or speed over time. When trying to identify these situations as those seen in graphing logarithmic functions, it is important to be able to recognize these graphs. It is also important to recognize graphs of exponential functions and their importance as the logarithmic inverse.

How to find the graph of a logarithmic equation with a base greater than one.

A short description of the graph of a logarithmic function.

Ex: Graph an Exponential Function and Logarithmic Function

This video explains how to graph an exponential and logarithmic function on the same coordinate plane. The two functions are inverses.

This video explains how to match exponential and logarithmic functions to graphs based upon the properties of the functions.

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