# Lesson 14

Solving Exponential Equations

• Let’s solve equations using logarithms.

### 14.1: A Valid Solution?

To solve the equation $$5 \boldcdot e^{3a} = 90$$, Lin wrote the following:

\displaystyle \begin {align}5 \boldcdot e^{3a} &= 90\\ e^{3a} &= 18\\ 3a &= \log_e 18\\ a&=\frac {\log_e 18}{3}\\ \end {align}

Is her solution valid? Be prepared to explain what she did in each step to support your answer.

### 14.2: Natural Logarithm

1. Complete the table with equivalent equations. The first row is completed for you.
exponential form logarithmic form
a. $$e^0= 1$$ $$\ln 1=0$$
b. $$e^1= e$$
c. $$e^\text{-1} = \frac{1}{e}$$
d.   $$\ln \frac{1}{e^2} = \text-2$$
e. $$e^x = 10$$
2. Solve each equation by expressing the solution using $$\ln$$ notation. Then, find the approximate value of the solution using the “ln” button on a calculator.
1. $$e^m=20$$
2. $$e^n=30$$
3. $$e^p=7.5$$

### 14.3: Solving Exponential Equations

Without using a calculator, solve each equation. It is expected that some solutions will be expressed using log notation. Be prepared to explain your reasoning.

1. $$10^x = 10,\!000$$
2. $$5 \boldcdot 10^x = 500$$
3. $$10^{(x+3)} = 10,\!000$$
4. $$10^{2x} = 10,\!000$$
5. $$10^x = 315$$
6. $$2 \boldcdot 10^x = 800$$
7. $$10^{(1.2x)} = 4,\!000$$
8. $$7\boldcdot 10^{(0.5x)} = 70$$
9. $$2 \boldcdot e^x=16$$
10. $$10 \boldcdot e^{3x}=250$$

1. Solve the equations $$10^{n} = 16$$ and $$10^{n} = 2$$. Express your answers as logarithms.
2. What is the relationship between these two solutions? Explain how you know.

### Summary

So far we have solved exponential equations by

• finding whole number powers of the base (for example, the solution of $$10^x = 100,\!000$$ is 5)
• estimation (for example, the solution of $$10^x = 300$$ is between 2 and 3)
• using a logarithm and approximating its value on a calculator (for example, the solution of $$10^x = 300$$ is $$\log 300 \approx 2.48$$)

Sometimes solving exponential equations takes additional reasoning. Here are a couple of examples.

\displaystyle \begin {align} 5 \boldcdot 10^x &= 45\\ 5 \boldcdot 10^x &= 45\\10^x &= 9\\x &=\log 9\\ \end {align}

\displaystyle \begin {align} 10^{(0.2t)} &= 1,\!000\\ 10^{(0.2t)} &= 10^3\\ 0.2t &= 3\\ t &= \frac {3}{0.2}\\ t &=15\\ \end {align}

In the first example, the power of 10 is multiplied by 5, so to find the value of $$x$$ that makes this equation true each side was divided by 5. From there, the equation was rewritten as a logarithm, giving an exact value for $$x$$.

In the second example, the expressions on each side of the equation were rewritten as powers of 10: $$10^{(0.2t)}=10^3$$. This means that the exponent $$0.2t$$ on one side and the 3 on the other side must be equal, and leads to a simpler expression to solve where we don't need to use a logarithm.

How do we solve an exponential equation with base $$e$$, such as $$e^x = 5$$? We can express the solution using the natural logarithm, the logarithm for base $$e$$. Natural logarithm is written as $$\ln$$, or sometimes as $$\log_e$$. Just like the equation $$10^2 =100$$ can be rewritten, in logarithmic form, as $$\log_{10}100 = 2$$, the equation $$e^0 = 1$$ and be rewritten as $$\ln 1 = 0$$. Similarly, $$e^{\text-2} = \frac{1}{e^2}$$ can be rewritten as $$\ln \frac{1}{e^2} = \text{-}2$$.

All this means that we can solve $$e^x = 5$$ by rewriting the equation as $$x = \ln 5$$. This says that $$x$$ is the exponent to which base $$e$$ is raised to equal 5.

To estimate the size of $$\ln 5$$, remember that $$e$$ is about 2.7. Because 5 is greater than $$e^1$$, this means that $$\ln 5$$ is greater than 1. $$e^2$$ is about $$(2.7)^2$$ or 7.3. Because 5 is less than $$e^2$$, this means that $$\ln 5$$ is less than 2. This suggests that $$\ln 5$$ is between 1 and 2. Using a calculator we can check that $$\ln 5 \approx 1.61$$.

### Glossary Entries

• $e$ (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.

• natural logarithm

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.