Solving Log and Exponential Equations

Solving Exponential Equations with Logs

Math 4 - 5/7/21

The most common logarithms used are base 10 and base e...

• e is a number. e is roughly 2.718. Click the "e" button on your calculator.

• For example, $\log_e\left(e\right)=1$  

• Instead of writing " $\log_e$  ", we write " $\ln$  " and it is called the "natural logarithm."

• Find the "LN" button on your calculator and try LN(e), LN(1), LN(5)

Base 10

• Notice the "LOG" button on your calculator? Try LOG(10), LOG(100) and LOG(0.01).

• It is common notation to simply write " $\log$  " without a base when we are referring to  $\log_{10}$  .

• So...  $\log_e\left(x\right)=\ln\left(x\right)$  and $\log_{10}\left(x\right)=\log\left(x\right)$  

• But other bases stays the same! This notation isn't only your calculator. If you type "log(1000)" into google, you will get 3 as an answer. If you type "ln(e)," you will get 1. 

Multiple Choice

$$\ln\left(4\right)\approx$$  

• 1.386

• 1.863

• 1

• 4

Multiple Choice

Without using your calculator... $$\log\left(100\right)=$$  

• 1
  

• 2

• 1.5

• 3

Multiple Choice

$$\ln\left(e\right)=$$  

Without using your calculator...

• 0.5

• 2.718

• 1

• 0

Multiple Choice

$$\ln\left(x\right)$$  means

• Natural logarithm of x

• Log base 10 of x

• ell enn of ex

• yes

We use the  $\ln$  and  $\log$  functions to solve exponential equations:

•  $2^x=12$  

•  $\log\left(2^x\right)=\log\left(12\right)$  

• Using the power property, this becomes    $x\cdot\log\left(2\right)=\log\left(12\right)$  

• Then divide both sides by  $\log\left(2\right)$  and you get 

•  $x=\frac{\log\left(12\right)}{\log\left(2\right)}$  

Multiple Choice

Solve $$3^x=15$$  

• x = 5

• x = 3.5

• x = 2.465

• x = 2.718

Multiple Choice

What is the value of x such that  $$4^x=8$$  ?

• 1.8

• 1.25

• 1.5

• 1.75