Exponent Equations with Different Bases

Exponents & Exponential Equations

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

​Dr. Tom Giles

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?

• $$-\frac{8}{7}$$  

• $$\frac{7}{5}$$  

• $$\frac{1}{2}$$  

• $$\frac{1}{10}$$  

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?