This activity extends students’ understanding of logarithms to include logarithms in another base. Students analyze patterns in a base-2 logarithm table and notice that it can be interpreted the same way as the base-10 table, except the values in the right-hand column are the exponents in expressions with a base of 2 (MP7).

Students use the table to evaluate base-2 log expressions and to solve simple exponential equations in base 2. They continue to reason abstractly and quantitatively (MP6) about the meaning of each parameter in the equations.

The work here prepares students to see an equation in the form of and one in the form of as equivalent, an understanding that they will develop in the next activity.

If students are not sure where to start when asked to solve the equations, consider saying:

• “Tell me more about the expressions in the Warm-up and what they mean.”

• “How does the expression relate to the exponent that 2 is raised to?”

Select previously identified students to share their responses and reasoning. Focus the discussion on two key ideas:

• In the equation , we know that the value of is 3 because , but we also know that the value of is , because by definition, is the exponent to which 2 should be raised to get 8. So we can express the solution as either 3, or , as the two are equal.
• When solving exponential equations, the solutions expressed in logarithms are exact solutions, whereas those written in decimals are approximations. For example, the solution to is or 3. In this case, both are exact solutions. For , the solution is exact, but the value in the base-2 log table, 2.3219, is not. If we raise 2 to the exponent of 2.3219, the result is not exactly 5.

In this activity, students make explicit connections between equivalent equations in exponential form and in logarithmic form. They also write equations to represent descriptions of exponential relationships. Working across different forms and representations reinforces students’ understanding of the meaning of logarithms. Along the way, they continue to attend carefully to the meaning of each parameter in the equations they write (MP6).

After converting a series of numerical equations from one form to the other, students generalize the equivalence of the two forms algebraically (MP8). Note that there are restrictions on the parameter : the base cannot be negative, and we don’t typically look at bases that are less than 1. Students don’t need to concern themselves with this at the moment, however.

The activity includes equations in various bases, but students should not be assessed on their readiness to work in bases other than 2 and 10.

If students do not yet correctly write the exponential and corresponding logarithmic equations in the table, consider asking:

• “Can you explain how you wrote your equation.”

• “What is the same and what is different about the equations  and ?”

Display the table, and invite students to share their solutions.

Make sure students understand the connections between the two forms. Consider annotating the parameters in the two forms of equations to help illustrate the connections:

Explain to students that base-10 logarithms are common enough that they are sometimes written without the subscript 10, and the base is assumed to be 10. For example, may be written as , which is equivalent to . In all other logarithms, the base is shown.