Exponential Equations

Exponential Equations

When the bases are the same and when bases are not the same.

Poll

How good are you at solving exponential equations?

• Excellent

• Pretty good

• Need Practice

• Completely lost

Multiple Choice

 $3^{\left(-3a-1\right)}=243$  

When looking at the equation above can 243 be expressed as base 3 to a power?

• Yes

• No

Multiple Choice

 $3^{\left(-3a-1\right)}=243$  

 $3^{\left(-3a-1\right)}=3^?$  

What is the exponent we are looking for?

• x

• 2

• -5

• 5

Multiple Choice

 $3^{\left(-3a-1\right)}=3^5$  
What should we do next?

• Take the logarithm of both sides.

• Set the exponents equal to each other.

Multiple Choice

 $3^{\left(-3a-1\right)}=3^5$  
 $-3a-1=5$  
What is the value of  $a$  ?

•  $6$  

•  $-2$  

•  $\frac{7}{4}$  

•  $-3$  

Multiple Choice

 $Solve\ the\ equation$  
 $5^{\left(-2a-3\right)}=25$  
 $a=?$  

•  $-\frac{5}{2}$  

•  $-\frac{1}{7}$  

•  $\frac{15}{8}$  

•  $\frac{3}{2}$  

Multiple Choice

 $4^{\left(x-1\right)}=64^{\left(2x-2\right)}$  
Which is the correct way to rewrite the equation using like bases?

•  $4^{\left(x-1\right)}=\left(4^2\right)^{\left(2x-2\right)}$  

•  $4^{\left(x-1\right)}=\left(4^3\right)^{\left(2x-2\right)}$  

Multiple Choice

Which is the correct way to solve the equation?

• 
  $4^{\left(x-1\right)}=\left(4^3\right)^{\left(2x-2\right)}$
  
  
  $4^{\left(x-1\right)}=4^{\left(6x-6\right)}$
  
  $x-1=6x-6$
  

• $4^{\left(x-1\right)}=\left(4^3\right)^{\left(2x-2\right)}$
  
  $4^{\left(x-1\right)}=4^{\left(6x-2\right)}$
  
  $x-1=6x-2$
  

Multiple Choice

 $4^{\left(x-1\right)}=\left(4^3\right)^{\left(2x-2\right)}$  

 $4^{\left(x-1\right)}=4^{\left(6x-6\right)}$  

 $x-1=6x-6$  
What is the value of x?

•  $-4$  

•  $1$  

•  $3$  

•  $-10$  

Multiple Choice

 $64^{\left(2n+1\right)}=16^{\left(2n+2\right)}$  
What is the value of n?

•  $-\frac{8}{7}$  

•  $\frac{7}{5}$  

•  $\frac{1}{2}$  

•  $\frac{1}{10}$  

Before we continue and use logarithms, lets consider logarithms in our life.

Multiple Choice

 $5^x=18$  
Can you get both sides of the equation with the same base?

• Yes

• No

Multiple Choice

 $5^x=18$  
Which is the correct way to proceed?

•  $\log\ 5^x=18$  

•  $\log\ 5^x=\log\ 18$  

Multiple Choice

 $5^x=18$  
 $\log5^x=\log18$  
What should be the next step?

• Divide both sides by log 5

• Divide both sides by log 18

• Use the Power Property of logarithms

Multiple Choice

 $5^x=18$  
 $\log5^x=\log\ 18$  
 $x\log5=\log18$  

 $\frac{x\log5}{\log5}=\frac{\log18}{\log5}$  
x=?

•  $2.9693$  

•  $1.7959$  

•  $1.2553$  

Multiple Choice

 $17^x=56$  
What is the value of x?

•  $1.7482$  

•  $4.0254$  

•  $1.4208$  

Open Ended

What is one or more things you would like to remember about today's lesson?

What is a question you still have and you need clarified?