# Lesson 14

Solving Exponential Equations

### Lesson Narrative

Prior to this point, students have solved a wide variety of equations, including using logarithms to solve simple exponential equations. In this lesson, they further use logarithms to solve equations that are increasingly more complex. They also learn that we can use logarithms to solve equations with base $$e$$, and that we refer to such a logarithm as the natural logarithm.

As they solve unfamiliar equations, students practice looking for and making use of structures from more familiar equations (MP7) and attending carefully to the parameters of the equations (MP6). Students also practice constructing logical arguments when they justify their or others’ solutions and explain why a certain value is a reasonable estimate for a given logarithm (MP3).

### Learning Goals

Teacher Facing

• Comprehend that the natural logarithm is used to express the solution to an exponential equation with base $e$.
• Use base 10 and natural logarithms to solve exponential equations.

### Student Facing

• Let’s solve equations using logarithms.

### Student Facing

• I can solve simple exponential equations using logarithms.

### CCSS Standards

• $e$ (mathematical constant)
The number $$e$$ is an irrational number with an infinite decimal expansion that starts $$2.71828182845\ .\ .\ .$$, which is used in finance and science as the base for an exponential function.
The natural logarithm of $$x$$, written $$\ln(x)$$, is the log to the base $$e$$ of $$x$$. So it is the number $$y$$ that makes the equation $$e^y = x$$ true.