# The logarithm function

## • Graph exponential and logarithmic functions, both by hand and using technology (F-IF.7e$^\star$, F-BF.B.5(+)).  • Verify that $f(x) = 10x$ and $g(x) = log_{10}(x)$ are inverses of one another (F-BF.B.4b(+)).

The previous sections focus on the definition of the logarithm as a notation for an unknown exponent and on solving equations involving exponentials and logarithms. In this section students start to view the logarithm as a function. This is analogous to the progression from learning about the square root symbol to studying the square root function. Students create graphs of logarithm functions by viewing them as the inverse of exponential functions, with inputs and outputs reversed. They apply their previous understanding of the nature of exponential functions to draw conclusions about the behavior of the graphs of logarithmic functions. For example, students who understand debt as an exponential function of time view the same situation as time being a logarithmic function of debt.

1 Exponentials and Logarithms II

WHAT: Students are asked to graph $f(g(x))$ and $g(f(x))$ where $f(x) = 10x$ and $g(x) = log_{10}(x)$ and find that they get the graph of $y = x$ each time, on different domains. Although they might initially use graphing technology, they should also provide reasons for their answer in terms of the definition of the logarithm (MP.3, MP.5).

WHY: The purpose of this task is to help students see that the definition of the logarithm as an exponent results in an inverse relationship between the exponential and logarithm functions with the same base. This prepares them to think about the shape of the graph of the logarithm function in terms of the shape of the graph of the exponential function.

2 Exponential Kiss

WHAT: These tasks form a series of explorations using Desmos for students to investigate the behavior of the graphs of logarithmic functions and to compare and contrast the graphs of exponential and logarithmic functions as inverses. Exponential Kiss shows a graphical non-calculus example where the number $e$ appears.

WHY: Graphing tools such as Desmos provide a low-barrier opportunity for students to explore and discuss these structures. Students have seen the base of the natural logarithm, $e$, and might wonder where it comes from. Although the most satisfying explanations arise from calculus, there are some examples where at least students can see that $e$ is a special value.

## External Resources

1 Earthquakes and Richter Magnitude

#### Description

WHAT: Northern Chile and Los Angeles, California have experienced recent earthquakes. These quakes are along the “Ring of Fire” where Earth’s underground plates often collide. In this lesson, students look for a pattern in the strength of earthquakes at different Richter magnitudes, graph the relationship and write an equation (F-LE.A.2$^\star$). The teacher can optionally extend the investigation to graphing on a log scale or articulating Richter magnitudes in terms of logarithms.

WHY: Richter magnitudes are logarithms of the energy expended by an earthquake (F-LE.B.5$^\star$). This activity gives students an opportunity to revisit linear versus exponential growth using their new understanding of logarithms.