Lesson 4 Common Sense Solidify Understanding

Learning Focus

Use logarithms to solve exponential equations.

Solve systems of equations that contain exponential functions.

How can we use logarithms and algebraic reasoning to help us solve exponential equations?

Open Up the Math: Launch, Explore, Discuss

You already know that our number system is base $10$ , so each of the different place values in a number are powers of $10$ . That makes the base $10$ exponential and base $10$ logarithmic functions very important. Because the base $10$ logarithm is so commonly used, it is called the “common log,” and to make the notation a little more compact, it is generally written without the base. So, it is used like this:

$\log 10 = 1$ , because $10^{1} = 10$

$\log 100 = 2$ , because $10^{2} = 100$

See how the base is just assumed to be $10$ when a base isn’t written? This is the kind of concise notation that mathematicians just love! When you use the log key on your calculator, it is automatically base $10$ . (Some technology allows you to enter a different base for a logarithm also.)

Each of the sections below contains puzzles for you to solve about base $10$ exponential functions. You will be using base $10$ logarithms and your knowledge of exponents to find missing values for exponential functions in tables, graphs, and equations. As you are working, watch for strategies that will help you to solve equations that have a variable in the exponent.

Table Puzzles

1.

Find the missing values of $x$ in the tables.

$x$	$y = 10^{x}$
$- 2$	$\frac{1}{100}$
$1$	$10$
	$50$
	$100$
$3$	$1000$

$x$	$y = 3 (10^{x})$
	$0.3$
$0$	$3$
	$94.87$
$2$	$300$
	$1503.56$

When you were trying to find $x$ when $y$ was $50$ in the first table, you were probably thinking of an equation like $50 = 10^{x}$ , even if you didn’t write it down. Here’s your chance! Write all the equations and their solutions that were used to find the missing $x$ in both tables. I gave you the first one. You’re welcome!

Table a: $50 = 10^{x}$

Table b:

2.

What strategy did you use to find the solutions to these equations when you were filling in the tables?

Graph Puzzles

3.

The graph of $10^{- x}$ is given below. Use the graph to solve the equations for $x$ and label the solutions.

a. $10^{- x} = 40$

$x =$

Label the solution with an $A$ on the graph.

b. $10^{- x} = 10$

$x =$

Label the solution with a $B$ on the graph.

c. $10^{- x} = 0.1$

$x =$

Label the solution with a $C$ on the graph.

4.

Let’s look a little closer at the solutions that you obtained from the graph. Consider the equation:

$10^{- x} = 10$

Would you get the same result if you took the base $10$ logarithm of both sides of the equation? Try it here:

$10^{- x} = 10$

$\log_{10} 10^{- x} = \log_{10} 10$

Keep going now by rewriting both sides, using logarithm properties.

5.

Let’s try it again with $10^{- x} = 0.1$ . Start by taking the base $10$ logarithm of both sides of the equation, then write equivalent expressions and check your answer with the graph.

$10^{- x} = 0.1$

6.

Why does this process give the same value as the graph?

7.

One of the equations you wrote in the table puzzles was: $94.87 = 3 (10^{x})$ .

How could you unwind this equation using basic operations and logarithms? Show your steps here.

Now you’re ready for the equation puzzles. Here we go!

Equation Puzzles

Solve each equation for $x$ .

8.

$10^{x} = 10, 000$

9.

$125 = 10^{x}$

10.

$10^{x + 2} = 347$

11.

$5 (10^{x + 2}) = 0.25$

12.

$10^{- x - 1} = \frac{1}{36}$

13.

$- (10^{x + 2}) = 16$

Combo Puzzles

Choose any method to solve.

14.

${\begin{matrix} y = 10^{x} \\ y = 100 \end{matrix}$

15.

${\begin{matrix} \begin{aligned} y = 10^{x} - 50 \\ y = 25 \end{aligned} \end{matrix}$

16.

${\begin{matrix} y = 10^{- x} \\ y = x + 5 \end{matrix}$

Ready for More?

Solve the equation $2^{x - 1} = 30$ using two methods:

Method 1: Use a base $2$ logarithm to find an exact expression for the solution.

Method 2: Use a base $10$ logarithm to find an approximate value for the solution.

Takeaways

Strategies for solving exponential equations and systems:

Vocabulary

common logarithm
Bold terms are new in this lesson.

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.

Retrieval

Calculate the interest earned based on the description, then create an equation that could be used for any number of periods of compounding given the same initial investment and percentage rate of increase.

1.

What will be the total value at the end of the year for an investment of $$ 1,000$ that has $1 %$ interest paid four times during the year?

Value at year’s end:

Equation to calculate for any amount of time, $t$ :

2.

If you are able to receive an $8 %$ return on your investment each year, how much will a $$ 5,000$ invest be worth after $10 years$ ?

Value after 10 years:

Equation to calculate for any amount of time, $t$ :