If student are overwhelmed by looking at all the numbers in the table, consider saying:

• “Tell me more about what you notice in each table.”

• “Where do you see patterns in these tables?”

The purpose of the discussion is to connect the values from a logarithm table to exponents.

Direct students’ attention to the reference created using Collect and Display. Ask students to share what the values in the table mean. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. 

Invite students to share how they went about using the table to solve for . To prompt them to make sense of the solution in each equation, discuss questions such as:

• “How does the table help you solve ?” (It tells us that 3 is the number to use for the exponent in for the expression to equal 1,000.)
• “The solution to is 0.9542. What does the solution mean?” (0.9542 is the number to use for the exponent in for it to equal 9.) “Does that solution make sense? How do we know?” (Yes. We know that and , so the exponent must be somewhere in between 0 and 1, likely close to 1, for the power of 10 to equal 9.)

Conclude the discussion by asking students how they could check to see if a solution found using the table is correct (We can enter it in a calculator as the exponent for base 10 and see if it produces the value on the other side of the equation). Show students how to do so by using a calculator to demonstrate that is approximately 90. If students wonder why it is not exactly 90, explain to students that the non-integer values in the table are all approximations rather than exact solutions. They will learn more about how to express the solutions exactly in future lessons.

If students do not yet correctly solve the equations, consider saying:

• “Tell me more about how you found the unknown exponents in the previous activity.” 

• “Refer to the tables from the previous activity. How could knowing that and help you determine the value of ?”

Invite students to share their conjectures on what the term logarithm means and what each part of a logarithmic equation tells us. Refine their definition and interpretation as needed. Make sure students understand that a logarithmic expression is the solution to an exponential equation; it is the exponent that makes the equation valid. In the examples here, the base is 10.

Explain to students that the expression is pronounced “the logarithm, base 10, of 1” or the “the log, base 10, of 1.” Help students generalize the relationship between an exponential expression and a log expression by writing:

Then explain that exponential and logarithmic equations can be rewritten in the other form in this way. Students will get additional practice rewriting equations in this way later in the unit.