Logarithmic Functions
Lessons in this chapter
About Logarithmic Functions
Logarithmic Functions covers logarithms from their definition through to real-world modeling.
Logarithmic Expressions introduces logarithms as the inverse operation to exponentiation, and Inverses of Exponential Functions confirms this by deriving the logarithm directly as an exponential function’s inverse.
Logarithmic Functions and Logarithmic Function Manipulation cover graphing logarithmic functions and using their properties to expand, combine, and simplify logarithmic expressions.
Exponential and Logarithmic Equations and Inequalities applies these tools to solving real equations, and Modeling with Logarithmic Functions and Semi-Log Plots close the chapter with logarithmic models and the semi-log plots used to reveal exponential patterns in real data.
All seven lessons open as practice worksheets in the page, from Logarithmic Expressions through Exponential and Logarithmic Equations and Inequalities to Semi-Log Plots. Each problem is scored as it is answered, which matters most where the logarithm properties are being applied two or three at a time. Free to try, with no login needed for your first 3.
Frequently asked questions
Logarithmic Expressions introduces the logarithm as the question an exponential answers, and Inverses of Exponential Functions derives it that way directly. Logarithmic Functions and Logarithmic Function Manipulation cover graphs and domains and the product, quotient, and power properties used to expand and combine expressions. Exponential and Logarithmic Equations and Inequalities applies them to solving, and Modeling with Logarithmic Functions and Semi-Log Plots close with real data, including the plot that turns exponential data into a straight line. This sits in Functions (F) in the high school Common Core, and in Unit 2 of the AP Precalculus course framework, Exponential and Logarithmic Functions.
A logarithm answers the question "what exponent gives this result": log_b(x) = y means b^y = x, making logarithms the inverse operation of exponentiation.
A logarithmic function's graph and its exponential inverse's graph are reflections of each other across the line y = x, since they undo each other.
The product, quotient, and power properties of logarithms let you expand a single complex logarithm into simpler pieces, or combine several logarithms into one.
Take the logarithm of both sides (matching bases when possible) to bring a variable down out of an exponent, then solve normally.
A semi-log plot graphs one axis on a logarithmic scale, turning an exponential pattern in the original data into a straight line that’s easier to model and read.