Properties of Logs-Basics

Properties of Logs

Properties and basic examples

Converting Logs and Exponentials

• Remember that logs and exponentials are inverses.

• Logs are exponents

• If  $\log_bM=y$  then it can be converted to  $b^y=M$  

• Be aware that  $b$  and $M$  must be  $>0$  and  $b\ne1$  

Multiple Choice

Logarithmic functions are the inverse of...

• Linear Functions
• Exponential Functions 
• Quadratic Functions 
• Polynomial Functions 

Converting examples:

•  $\log_28=x$  converts to  $2^x=8$  

• We now can evaluate to find:  $x=3$ 

• Recall that ( $\ln$ ) is the Natural log and has a base of  $e$  

•  $\log$  with no base written is the common log which has a base of 10

Multiple Choice

Rewrite log_vn = a in exponential form.

• v^a = n
• n^a = v
• vⁿ = a
• aⁿ = v

Multiple Choice

Change y = a^x into log form.

• x = log_ay

• y = log_ax

• x = log_ya

• y = log a

Multiple Choice

 What is $\log10^4$ ? (Hint: change into expon. form first)

• 10

• 4

•  $10^4$  

• 1

Multiple Choice

What is $\ln e^2$ ?  (Hint: what is the hidden base of ln?)

• 1

• e

• 2

•  $e^2$  

Exponent Rules

Since logs are exponents they follow many of the rules of exponents

There are 3 Log Properties

Product Property

Echoes the multiplication rule of exponents

A product of 2 expressions within a log can be expanded into a sum of those expressions

Multiple Choice

Expand: log(4x)

• log4-logx

• log4+logx

• 4logx

• xlog4

Multiple Choice

Rewrite as a single logarithm:

 $\log3\ +\ \log7$  

• log 10

• log 21

• log 3/7

• log 3/log 7

Multiple Choice

Expand  $\log_4\left(3xy\right)$  

•  $\log_4\left(3\right)\ +\ \log_4\left(x\right)-\ \log_4\left(y\right)$ 

•  $3\log_4\left(x\right)\ +\ 3\log_4\left(y\right)$ 

•  $\log_4\left(3\right)\ +\ \log_4\left(x\right)+\ \log_4\left(y\right)\$ 

•  $4\log\left(3\right)\ +\ 4\log\left(x\right)+\ 4\log\left(y\right)\$ 

Multiple Choice

Condense into a single logarithm.
 $\log_25+\log_29+\log_2w$  

•  $\log_2\left(14w\right)$  
  

•  $\log_2\left(45w\right)$  
  

•  $\log_2\left(\frac{14}{w}\right)$   
  

•  $\log_2\left(14+w\right)$  

Quotient Property

Echoes the Division Rule of Exponents

When 2 expressions within a log are divided they can be expanded into a difference of those expressions

Quotient Examples

Notice when condensing we write only ONE log term!

Multiple Choice

Expand:   $\log\left(\frac{x}{y}\right)$  

• logx+logy

• xlogy

• log(x-y)

• logx-logy

Multiple Choice

Expand
 $\log_6\left(\frac{y}{36}\right)$  

•  $\log_6y+\log_636$  

•  $\log_6y-\log_636$  

•  $\log_636+\log_6y$   

•  $\log_636-\log_6y$  

Multiple Choice

Condense 

 $\log_5y-\log_525$  

•  $\log_5\left(y-25\right)$  

•  $\log_5\left(25y\right)$  

•  $\log_5\left(\frac{y}{25}\right)$  

•  $\log_5\left(\frac{25}{y}\right)$  

Power Property

Power Property

In the example, the exponent of 3 can be brought down in front of the expression. We will use this property frequently when solving expon. equations.

Multiple Select

Which are equivalent to:

 $\log_24^2$ 

(There is more than 1 correct answer) 

•  $4\log_22$  

•  $2\log_42$  

•  $2\log_24$   

• 4

• 2

Multiple Select

Which are equivalent to:

 $2\log_39$  
(There is more than 1 correct answer)

•  $\log_39^2$  

•  $\log_32^9$  

• 4

• 3

• 9

Condensing/Expanding when more than 1 property is needed:

• Follow the order of operations

• When condensing, always do the Power Property first

Multiple Choice

Condense into one log:   $2\log_3\left(x\right)+\log_3\left(y\right)$  

Remember to use the power property first!

•  $\log_3\left(2xy\right)\$ 

•  $\log_3\left(x^2y\right)$ 

•  $\log_3\left(xy\right)^2$ 

•  $2\log_3\left(xy\right)$ 

Multiple Choice

Condense into a single log expression.

Multiple Choice

Expand:   this has a mix of Product and Quotient Prop.:

 $\log\left(\frac{8x}{yz}\right)\$  

• log8+logx-logy-logz

• log8+logx-logy+logz

• log(8x)-log(yz)

• 8logx-ylogz

Multiple Choice

Expand:    $\log\left(\frac{m^3}{n}\right)$  

• logm-log3-logn

• 3logm+logn

• 3logm-logn

• 3log(m-n)

In this Lesson you learned....

• How to convert between log and exponential forms

• The 3 Log properties: Product, Quotient and Power

• How to use the properties to expand/condense log expressions