Algebra/Logarithms

Logarithms (commonly called "logs") are a specific instance of a function being used for everyday use. Logarithms are used commonly to measure earthquakes, distances of stars, economics, and throughout the scientific world.

LogarithmsEdit

In order to understand logs, we need to review exponential equations. Answer the following problems:

1 What is 4 to the power of 3?

2 What is 3 to the power of 4?

3

 

4

 


Just like there is a way to say and write "4 to the power of 3" or " , there is a specific way to say and write logarithms.

For example, "4 to the power of 3 equals 64" can be written as:  

However, it can also be written as:

 

Once, you remember that the base of the exponent is the number being raised to a power and that the base of the logarithm is the subscript after the log, the rest falls into place. I like to draw an arrow (either mentally or physically) from the base, to the exponent, to the product when changing from logarithmic form to exponential form. So visually or mentally I would go from 2 to 5 to 32 in the logarithmic example which (once I add the conventions) gives us:  

So, when you are given a logarithm to solve, just remember how to convert it to an exponential equation. Here are some practice problems, the answers are at the bottom.

Properties of LogarithmsEdit

The following properties derive from the definition of logarithm.

Basic propertiesEdit

For all real numbers   with  , we have

  1.  
  2.  
  3.  
  4.  
  5.   (change of base rule).

ProofEdit

Let us take the log to base d of both sides of the equation  :

 .

Next, notice that the left side of this equation is the same as that in property number 1 above. Let us apply this property:

 

Isolating c on the left side gives

 

Finally, since  

 

ExamplesEdit

This rule allows us to evaluate logs to a base other than e or 10 on a calculator. For example,

 

Solve these logarithms

1

 

2

 

3

 

4 Evaluate with a calculator (to 5dp)

 

Find the y value of these logarithms

5  

y=

6  

y=

7  

y=


More propertiesEdit

Logarithms are the reverse of exponential functions, just as division is the reverse of multiplication. For example, just as we have

 

and

 

we also have

 

and

 

More generally, if  , then  . Also, if  , then  , so if the two equations are graphed, each one is the reflection of the other over the line  . (In both equations, a is called the base.)

As a result,   and  .

Common bases for logarithms are the base of 10 (  is known as the common logarithm) and the base e (  is known as the natural logarithm), where e = 2.71828182846...

Natural logs are usually written as   or   (ln is short for natural logarithm in Latin), and sometimes as   or  . Parenthesized forms are recommended when x is a mathematical expression (e.g.,  ).

Logarithms are commonly abbreviated as logs.

AmbiguityEdit

The notation   may refer to either   or  , depending on the country and the context. For example, in English-speaking schools,   usually means  , whereas it means   in Italian- and French-speaking schools or to English-speaking number theorists. Consequently, this notation should only be used when the context is clear.

Properties of LogarithmsEdit

  1.  
  2.  
  3.  

Proof:
 

 

  and  

  and  

 

 

 

and replace b and c (as above)

 

Change of Base FormulaEdit

  where a is any positive number, distinct from 1. Generally, a is either 10 (for common logs) or e (for natural logs).

Proof:
 

 

Put both sides to  

 

 

 

Replace   from first line

 

Swap of Base and Exponent FormulaEdit

  where a or c must not be equal to 1.

Proof:

  by the change of base formula above.

Note that  . Then

  can be rewritten as

  or by the exponential rule as

 

using the inverse rule noted above, this is equal to

 

and by the change of base formula