Lesson 14

This warm-up gives students a chance to see and analyze a worked solution to an exponential equation before they solve some in the next activity. An exponential equation with base \(e\) was chosen purposefully to reinforce that \(e\) is just a number, and, as such, students can work with it the same way they have worked with equations of other bases previously.

Students should be familiar with the first and last steps in the worked solution, but the second step may give them pause because the exponent is a product (rather than just a variable or a number). Monitor for students who recognize that the product could be treated as a single object and who see the middle two lines as equivalent equations in exponential and logarithmic forms.

Give students a moment of quiet think time, and then ask them to share their thinking with a partner.

Select students to explain Lin’s worked solution. If students struggle to explain the transition from \(e^{3a} = 18\) to \(3a = \log_e 18\), consider temporarily replacing the \(3a\) with a single variable, such as \(y\), and then ask students to interpret \(e^y = 18\) to \(y = \log_e 18\). Once students see these two equations as equivalent, given the definition of logarithm, then return to \(3a = \log_e 18\) and the division of each side by 3.

Some students may think that the equation is not completely solved because the logarithm is not evaluated. Remind students that a solution written in log notation is the exact solution, and that evaluating it would give us an approximation (a less-precise solution).

Building from the warm-up, in this activity students encounter more exponential equations with base \(e\). They learn that a logarithm with base \(e\) has a special name, the natural logarithm, and use the “ln” function on a calculator to estimate its value for different inputs.

To write equivalent equations in exponential form and logarithmic form, students look for and make use structures similar to those they previously encountered (MP7).

Display the table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students brief quiet think time, and then invite students to share their observations.

Explain to students that the logarithm of an exponential expression with base \(e\) is called the natural logarithm, written as \(\ln\). Emphasize that \(\ln x\) means the same thing as \(\log_e x\).

Once students have completed the table, ask them to pause. Make sure students recognize that each pair of equations can be interpreted the same way we interpreted equations in other bases, and that the notation \(\ln \text{(some number)}\) can be thought of as the exponent to which we raise a base \(e\) to produce that number.

Give students access to a scientific calculator and ask them to locate the “e” and “ln” buttons on the calculator. Ask them to try evaluating \(\ln e\) and \(\ln 10\) and verify that they get 1 and approximately 2.302585.

Students may struggle getting started with the presence of the number \(e\). Ask them to look at the first row of the table. The reason \(\ln(1) = 0\) is that 0 is the exponent of \(e\) that gives 1. Ask, for \(\ln(e)\), what power of \(e\) gives a result of \(e\)? (1, since \(e^1=e\).)

Invite students to share their solutions for the 3 unknown values. Resolve any discrepancies, such as any disagreements about approximate values determined using technology.

In this activity, students use logarithmic expressions to solve exponential equations. A variety of problems are chosen so that some can be solved through exponential reasoning, while others, in order to give an exact answer, require the use of logarithms.

Monitor for students who use the following strategies to solve the given equations:

• Reference the base 10 log table from an earlier lesson (for example: to find \(10^x = 10,\!000\), look to the last entry in the table, which gives the value for \(\log 10,\!000\)).
• Use exponential reasoning (for example: by thinking of 10,000 as \(10^4\) and reason that if \(10^{2x} = 10^4\), then \(2x = 4\) and \(x = 2\)).
• Use the definition of logarithm to rewrite equations (for example: if \(10^n = 315\), then \(n\) is the exponent to which 10 needs to be raised in order to give 315, so \(n = \log 315\)).

For students who struggle solving the equations explicitly or in logarithmic form, recommend trying to isolate a power of 10 on one side (with the variable). Sometimes the other side can be written as a power of 10 and sometimes it cannot. When it can, this can be used to solve for the variable. When it cannot suggest that they think about the meaning of the base 10 logarithm and what it tells them.

Focus the discussion on the different strategies students used to solve the equations. Invite previously identified students to share in the order given in the Activity Narrative.

During the discussion, help students connect the last few questions to the discussions in earlier lessons about exact solutions versus approximations. Also highlight the fact that the solution to several of these problems (for example, \(10^n = 315\) or any of the problems involving \(e\)) can only be expressed exactly using logarithms because neither the log table nor exponential reasoning will produce an exact value. On the other hand, using logarithms to solve the first few problems is not necessary. Rules of exponents are well suited for solving these problems.