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 was chosen purposefully to reinforce that 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.

Select students to explain the worked-out solution. If students struggle to explain the transition from to , consider temporarily replacing the with a single variable, such as , and then ask students to interpret to . Once students see these two equations as equivalent, given the definition of logarithm, then return to 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).

In this activity students encounter more exponential equations with base . They learn that a logarithm with base 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 of structures similar to those they previously encountered (MP7).

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 these strategies to solve the given equations. The strategies are listed in order of flexibility.

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