Unit 2 Logarithmic Functions

Lesson 1

Learning Focus

Find exact and estimated values of logarithms.

Make conjectures about logarithm properties.

Lesson Summary

In this lesson, we found exact and approximate values for logarithmic expressions. We drew conclusions about properties of logarithms that are always true. We also identified characteristics of logarithmic functions that will help us to graph logarithms in the next lesson.

Lesson 2

Learning Focus

Identify common features of the graphs of logarithmic functions.

Transform the graph of a logarithmic function.

Lesson Summary

In this lesson, we graphed logarithmic functions by hand and used technology to determine common features in the graphs. We used technology to identify how the transformations appear in the equations of logarithmic functions. We examined graphs and wrote equations for the transformed functions, and we graphed transformed functions given the equations.

Lesson 3

Learning Focus

Use graphs to discover properties of logarithms.

Justify conjectures about logarithms.

Lesson Summary

In this lesson, we examined graphs to find equivalent expressions for logarithmic functions. We justified three logarithm properties that are true for any logarithm base. They are:

Logarithm of a Product Rule: $\log_{b} (x y) = \log_{b} x + \log_{b} y$

Logarithm of a Quotient Rule: $\log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y$

Logarithm of a Power Rule: $\log_{b} (x^{k}) = k \log_{b} x$

Lesson 4

Learning Focus

Use logarithm properties to find equivalent algebraic expressions.

Use logarithm properties to find values for logarithmic expressions using other known values.

Lesson Summary

In this lesson, we learned to find the value of a logarithmic expression combining known values and using the logarithm properties. We determined if certain logarithmic equations were true and reinforced that the logarithm product and quotient rules are only for multiplication and division inside the argument of the logarithm.

Lesson 5

Learning Focus

Use logarithms to solve exponential equations.

Solve systems of equations that contain exponential functions.

Lesson Summary

In this lesson, we found solutions to base $10$ exponential equations by using base $10$ logarithms to undo the equation. We used tables and graphs to help support our thinking about whether the solutions we find are reasonable. We solved systems of equations both by finding the intersection of the graphs and by finding the value that makes both equations true.

Lesson 6

Learning Focus

Understand how different factors affect the amount earned from compound interest.

Determine the features of a new function: $f (x) = e^{x}$ .

Lesson Summary

In this lesson, we explored the effects of changing the number of compounding periods in an investment that earns compound interest. In the course of this exploration, we found the number $e$ , an irrational number that arises in many exponential growth and decay contexts. We investigated the function $f (x) = e^{x}$ and identified the features, finding that the behavior of the function is very similar to $y = 2^{x}$ , a function with which we have a lot of experience.

Lesson 7

Learning Focus

Model continuous growth and decay using base $e$ exponential functions.

Solve exponential equations using natural logarithms.

Lesson Summary

In this lesson, we modeled continuous growth and decay using the formula that is a base $e$ exponential function. We learned to use the base $e$ logarithm, called the natural log, to solve exponential equations when the variable is in the exponent. The natural logarithm is written, $\ln x$ , and has a dedicated function on most calculators.