Mastery Check Solutions

Show What You Know

Two groups of students solved the same equation and found the same incorrect answer. They argued that because they found the same result, both groups must be correct.
Find and correct the error that each group made.

$\begin{array}{rcl} 2 e^{5 x} - 4 & = & 6 \\ \frac{2 e^{5 x}}{2} & = & \frac{10}{2} \\ e^{5 x} & = & 5 \\ \ln e^{5 x} & = & \ln 5 \\ \frac{5 x}{5} & = & \frac{\ln 5}{5} \\ x & = & \frac{\ln 5}{5} \approx 0.3219 \end{array}$

Note

To help with the next part of the problem, indicate the step at which the error occurs.

Explain the error that each group made and how to correct their misunderstanding.

Sample:

Green error: $\ln 5$ is irrational, and 5 is rational. Therefore, 5 cannot be simplified out of the problem. You must use technology to find the decimal equivalent.

Purple error: You cannot simplify an exponent, 5x, and a base, 5. Instead, you must rewrite the problem with a natural log and use technology to find the decimal equivalent.

Note

Remember, most logarithms are irrational numbers (numbers that cannot be written as a ratio). You cannot simplify a rational and an irrational number to one term, but you can find a decimal equivalent with technology.

Q: What is an irrational number?

A: A number that cannot be written as a ratio or fraction

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Write equivalent expressions using natural logarithms and the number e.
Use the properties of logarithms to simplify or evaluate expressions with natural logarithms and the number e.
Solve natural logarithm equations.