Exponential & logarithmic functions

This course is a part of Algebra (all content), a 20-course Topic series from Khan Academy.

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

This course contains 31 segments:

Exponent properties review

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

Radicals

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.

Rational exponents intro

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

Properties of exponents (rational exponents)

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

Advanced exponent & radical evaluation

Evaluate elaborate expressions that contain radicals and fractional exponents.

Simplifying square roots

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.

Simplifying radicals (higher-index roots)

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.

Radicals (miscellaneous videos)

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

Exponential vs. linear growth

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

Exponential expressions

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

Graphs of exponential growth

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.

Exponential vs. linear growth over time

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

Exponential decay

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

Exponential functions from tables & graphs

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

Equivalent forms of exponential expressions

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

Solving exponential equations using properties of exponents

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.

Introduction to rate of exponential growth and decay

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

Interpreting the rate of change of exponential models (Algebra 2 level)

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

Constructing exponential models according to rate of change (Algebra 2 level)

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

Advanced interpretation of exponential models (Algebra 2 level)

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.

Distinguishing between linear and exponential growth (Algebra 2 level)

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

Introduction to logarithms (Algebra 2 level)

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.

The constant e and the natural logarithm (Algebra 2 level)

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

Properties of logarithms (Algebra 2 level)

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!).

The change of base formula for logarithms (Algebra 2 level)

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.

Logarithmic equations (Algebra 2 level)

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

Solving exponential equations with logarithms (Algebra 2 level)

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

Solving exponential models (Algebra 2 level)

Learn how to solve word problems that require exponential equations.

Graphs of exponential functions (Algebra 2 level)

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

Graphs of logarithmic functions (Algebra 2 level)

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

Logarithmic scale (Algebra 2 level)

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