# Understand the definition of a logarithm

## • Understand the definition of a logarithm as the solution to an exponential equation (F-LE.A.4$^\star$).  • Practice evaluating log expressions and converting between the exponential form of an equation and the logarithmic form (F-LE.A.4$^\star$).

The logarithm can be defined operationally. Just as $\sqrt[3]{2}$ is defined to be the positive real number that when multiplied by itself three times is equal to $2$, the solution to $2^x=k$ is defined to be $x=\log_2(k)$. In this section students develop this definition through an exploratory activity. They analyze a number of true statements about logarithms without having been told the meaning of the notation, make conjectures about the pattern they fit using their knowledge of exponents, and express their meaning in terms of an equivalent exponential equation. They come up with a definition of the logarithm by precisely describing what they see and generalizing it. They then practice interpreting and converting logarithmic expressions.

## External Resources

1 Introducing Logs

#### Description

WHAT: In the Google document linked to this blog post, students are introduced to logarithms using a puzzle in which they are given several statements of the form $\log_2(8)=3$ and then asked to fill in the blanks in several more such statements, using their knowledge of powers of whole numbers. They are asked to write a definition of what the equation $\log_b(a)=x$ means (MP.6) and they generate some puzzles of their own. Finally, they are asked to write an expression they would enter into a calculator to find a missing exponent (F-LE.A.4$^\star$, MP.8).

WHY: The purpose of this activity is to help students understand the logarithm as a notation for a missing exponent. It is intended to be given to students before they have had a formal introduction to logarithms.

2 Log War

#### Description

WHAT: This blog post provides a document that can be cut up to create a special decks of cards. Each card displays a different logarithmic expression, for example, $\log_3(1/3)$. Students (in pairs) play the familiar card game, War, deciding which card wins by mentally evaluating the logarithmic expression, which includes being able to evaluate expressions with fractional exponents (N-RN.A.2).

WHY: The purpose of this activity is to give students practice with logarithm notation so that they can internalize its meaning, including becoming familiar with the convention that logarithms written without a base are understood to be logarithms with base 10 (F-LE.A.4$^\star$).