# Lesson 9

What is a Logarithm?

### Lesson Narrative

This lesson introduces students to logarithms. They see that a logarithm can be used to express the exponent of an expression written in a certain base and having a certain value. For example, if the expression $$10^y$$ has a value of 50, we can estimate the value of $$y$$ to be about 1.69897, but we can also write $$\log_{10} 50$$ to express the value of $$y$$. In other words, a logarithm can be defined as the solution to an exponential equation such as $$10^y=50$$.

Students begin by making sense of the values in a base 10 logarithm table, looking for patterns, and using their observations to solve exponential equations in base 10 (MP8). They recognize some whole-number log values as the exponents that produce certain powers of 10 and verify that this is also the case for decimal log values in the table (by entering those values as the exponent for $$10^x$$ in a calculator).

Then, students encounter logarithmic equations for the first time. They analyze and interpret the parameters in the equations, and work to articulate the meaning of each part in the equation precisely (MP6). All exponential and logarithmic equations in this lesson are in base 10. In future lessons, students will encounter equations in other bases.

Note that in this lesson and the following, students are learning to make sense of logarithms, so use of technology to evaluate logarithms is not recommended. In later lessons, students will have opportunities to use a calculator to find log values.

### Learning Goals

Teacher Facing

• Comprehend the definition of a logarithm as the solution to an exponential equation.
• Identify the parameters in expressions involving a logarithm.

### Student Facing

• I understand that a logarithm is a way to represent an exponent in an exponential equation.

Building Towards

### Glossary Entries

• logarithm

The logarithm to base 10 of a number $$x$$, written $$\log_{10}(x)$$, is the exponent you raise 10 to get $$x$$, so it is the number $$y$$ that makes the equation $$10^y = x$$ true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. $$\log_2(x)$$ is the number $$y$$ that makes the equation $$2^y = x$$ true. The logarithm to the base $$e$$ is called the natural logarithm, and is written $$\ln(x)$$.