# Exponential and Logarithmic Equations

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## Objective

Students will be able to solve exponential and logarithmic equations

#### Big Idea

Here is one of those places that we get to show students how logarithms are useful.

## Warm Up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Exponential and Logarithmic Equations which asks students to estimate the value of several logarithms.

I also use this time to correct and record the previous day's Homework.

## Why Are Logarithms Necessary

30 minutes

The central goal of this lesson is for students to be able to solve equations in logarithmic or exponential form.  We look back to the zombie invasion scenario done several lessons ago.   I use this scenario to show students the need to be able to find non-rational logarithms.  Using the function f(x)=2x , or each zombie infecting one person per day, the students are asked to figure out how many zombies there will be in 10 days.

Next, they are asked to figure out how many days it will take to convert the whole town (Math Practice 4).   Make sure you have the population of your town handy for this problem.  This is the lead-in to logarithms being necessary for the solution to some exponential equations.  I give the students some time to work on this problem and then we discuss this as a class.  The goal here is for the class to come up with the idea of using logarithms to find the solution to the final problem on their own.

Next,  I ask the students how to find the solution to log2(the population of your town).  We look at estimating logarithms as well as the fact that log152 is the same as log10152 .  Estimating things like logs and roots is a really great way to increase student understanding of those concepts.  We talk about whether this estimate will be enough.   Some may be pretty close through guess and check but we need to be more accurate.

## Change of Base Theorem

40 minutes

Next, the students are introduced to the Change of Base Theorem.  My department has decided to save the laws of logarithms for a post-Algebra 2 course so we won't develop this conceptually until next year.  For more information on why the change of base works, please watch this video.

## Equations with Logarithms

5 minutes

The students now use their new knowledge to simplify non-standard logarithms and logarithmic equations.  I run this as a Guided Practice

Please see the PowerPoint for detailed presentation notes.

## Homework

1 minutes

This Homework has a couple of straight forward equations to solve and then it goes into some conceptual questions.  It is really important to train students to answer problems that request explanations with complete sentences.  The assignment for this section asks students to solve a variety of logarithmic and exponential equations to practice the skills they learned in class.  It also asks the students to analyze the logarithm equation logx2=0 (Math Practice 2).  The goal of this equation is to realize that a logarithm that isn't in the form logx1=0 can never equal zero.  They are also asked to identify that the answer to logx1=0 is all real numbers.  Finally, they return to the zombie scenario, write a function and answer some questions based on that function.

## Exit Ticket

5 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

This Exit Ticket asks students to solve one exponent and one logarithm equation.