# Exponential & logarithmic functions

Contents

This topic covers:
- Radicals & rational exponents
- Graphs & end behavior of exponential functions
- Manipulating exponential expressions using exponent properties
- Exponential growth & decay
- Modeling with exponential functions
- Solving exponential equations
- Logarithm properties
- Solving logarithmic equations
- Graphing logarithmic functions
- Logarithmic scale

44 exercises available

Practice using the exponent properties to rewrite powers, where the exponent is an integer that can either be positive or negative.

Radicals (also known as roots) are a generalization of square roots. They are the inverse operation of any power. For example, the 5th root of 32 is 2, because 2⁵=32.

Learn the definition for raising a number by a fractional exponent, like 8^⅔.

In this tutorial you will rewrite variable expressions with rational exponents using the properties of exponents.

Evaluate elaborate expressions that contain radicals and fractional exponents.

We know how to evaluate square roots of perfect squares. For example, √16=4. What about the other square roots? It's harder to give an exact number, but we can simplify them so we have a better understanding of their value. For example, √32=4⋅√2. Learn more about it in this tutorial.

Just like we can simplify square roots, we can simplify other radicals. This tutorial covers the simplification of higher-index roots. For example, ∜48=2⋅∜3.

Various videos we have that cover advanced/special skills related to radical expressions.

Learn about exponential growth and decay, and specifically how it always grows (or diminishes) by equal factors.

Learn how an exponential function behaves as the value of its input increases to positive infinity or decreases to negative infinity. Learn how to graph basic exponential functions.

Learn how to analyze the formulas of basic exponential functions in order to find their common ratio, initial value, and other parameters.

Learn how to analyze the graphs, or tables of values, of basic exponential functions in order to find their common ratio, initial value, and other parameters.

Learn how to construct exponential functions to model real-world situations.

Learn how to construct exponential functions and then analyze them to model and solve real-world problems.

Learn how distinguish between linear and exponential growth, and learn the differences between the end behavior of polynomial and exponential functions.

Learn how to manipulate exponential expressions in different ways. For example, rewrite 8^x as 2^(3x).

Learn how to solve advanced exponential equations by manipulating the expressions in the equations using the properties of exponents. For example, solve 2^(x+1)=8^x by rewriting 8^x as 2^(3x) and then equating x+1=3x.

Learn about different ways of describing the rate of change of exponential functions.

Learn how to analyze exponential modeling functions in order to find their rate of change.

Learn how to find the modeling function of an exponential real world context, according to the description of its rate of change.

Learn how to interpret an exponential modeling function by first manipulating it according to your needs. For example, rewrite 5^(2x+1) as 5*25^x to find that the unit growth factor is 25.

Learn how to analyze real-world quantitative relationships given as tables of values to determine whether they represent linear growth or exponential growth.

Learn how we define logarithms and use this definition in order to evaluate various logarithms. For example, evaluate log_2(8) as 3 by realizing that 2^3=8.

Learn about a very special constant in math that has a pivotal role in the world of exponential and logarithmic function, the constant e.

Learn about special properties of logarithms that help us rewrite logarithmic expressions in different equivalent (much like we use properties of exponents to rewrite exponential expressions!).

Learn how to rewrite any logarithmic expression in a different base. For example, rewrite log_2(3) as ln(3)/ln(2). This is very helpful for evaluating logarithms with a calculator, which only evaluates base-10 and base-e logarithms.

Learn how to solve equations that contain logarithmic expressions. For example, solve log(x)+log(3)=log(7).

Learn how to solve any exponential equation by using logarithms. For example, solve 2^x=3 by calculating log_2(3).

Learn how to solve word problems that require exponential equations.

Learn about the graphs of advanced exponential functions of the form y=a*b^(x+c)+d.

Learn about the graphs of logarithmic functions, and how they relate to graphs of exponential functions.

Enrich your knowledge with some videos about logarithmic scales and how useful they are.