Logarithmic functions go hand in hand with exponential functions. Many times there may be a problem involving exponents that is difficult to solve. Logarithms, otherwise known as Logs, will help to solve these types of problems. First understand what a Logarithmic function is. Logarithmic functions can be written in the following form.

log_ax = b

 

The a is called the base. Without going into a long drawn out explanation of the log function and its relationship to exponents, here is the relationship between log form and exponents.

 

log_ax = b

 

means

 

a^b = x

 

 

 

 

 

Example

 

Evaluate log₄64

 

To understand this easier, set it up as an equation by setting it equal to x.

 

log₄64 = x

 

Apply the relationship just discussed.

 

4^x= 64

 

This can be understood as “4 to what power is 64?”

 

x = 3

 

Because 4³= 64.

 

3

 

Answer!

 

 

 

 

 

 

Example

Evaluate log₃9

 

Set it equal to x.

log₃9 = x

 

Apply the relationship.

3^x= 9

 

Ask “3 to what power is 9?”

x = 2

 

Answer!

 

 

 

 

 

 

 

Example

Evaluate log₂₅5

 

Set it equal to x.

Log₂₅5 = x

 

Apply the relationship.

 

25^x= 5

 

Ask “25 to what power is 5?” In this case the square root of 25 is 5 so that means that 25 is being raised to the ˝ power.

 

x = 1/2

 

1/2

 

Answer!

 

 

 

 

There are certain properties of logs that can be applied at different times to simplify work.

 

log_x1 = 0

 

This is true because x⁰ = 1. The x can be anything, and anything to the zeroth power is 1.

 

 

 

log_xx = 1

 

This is true because x¹= x. Anything raised to the first power is itself.

 

 

 

log_xx^y = y

 

Applying the relationship between logs and exponents to the left side gets x^y= x^y.

 

 

 

The most familiar base to work with is 10. It is easy to calculate the log of any number with a base of 10. On a scientific calculator find the button that says “log”, this calculates the log of any number in the base of 10. When not in the base of 10 do not use this button. It will be explained later what to do when there is a case like that.

 

When compounding continuously, use the e button on a calculator. In dealing with logs there is something that relates directly to that, it is the natural log. Natural logs are written using the symbol ln. Here is the relationship between logs and natural logs.

 

log_ex = ln x

 

Natural logs are used whenever there is a base of e. Like the logs of base 10, it is easy to calculate a natural log with a calculator. Look for the button with “ln” on it. This button is used to calculate the natural log of any number.

 

 

 

Example

 

Evaluate log_e5

 

Rewrite as a natural log since it has a base of e.

ln 5

 

Use a calculator.

 

1.609                                                                                                                               

 

Answer!

 

 

 

 

 

 

 

Example

Evaluate log_e

 

 

Rewrite as a natural log.

ln

 

 

Use a calculator.

 

-.693

 

Answer!

 

 

 

 

 

 

Natural logs have properties of their own.

ln 1 = 0

ln e = 1

ln e^x = x

 

So far the calculator has been used to solve problems involving logs of base 10 and base e (natural logs). Here is how to change the base of a problem to make it base 10, to make it possible to use a calculator to solve it. Following is the general formula to change the base of any log function. Notice that the base is not specified. This means use whatever base is easiest as long as it is the same in the top and the bottom. Most of the time use base 10, since that can be easily calculated on a calculator. Occasionally, it might be easier to use base e, since a calculator can find the natural log also.

 

 

 

 

This formula is also referred to as “changing the base”.

 

 

 

 

 

Example

Find log₃5

 

Change the base.

 

 

Since there is a base of 10, use a calculator to find the log 5 and log 3.

 

 

Divide.

 

1.46497

 

Answer!

 

 

 

 

 

Example

 

Find log₄120

 

Change the base.

 

 

 

Since there is a base of 10, use a calculator to find the log 120 and log 4.

 

 

Divide.

 

3.45345

 

Answer!

 

 

 

 

 

 

There are some rules of logarithms that allow simplification of problems.

 

Multiplication

log_abc = log_ab + log_ac

 

Division

log_a= log_ab - log_ac

 

Exponential

log_ab^c = c log_ab

 

 

 

Example

Rewrite log₃5x

 

Use the rule of multiplication.

log₃5 + log₃x

 

Answer!

 

 

 

 

 

 

Example

Rewrite log₇x² y

 

First, use the rule of multiplication.

 

log₇ x² + log₇y

 

Now use the rule of exponents on the first term.

 

2log₇ x + log₇y

 

Answer!

 

 

 

 

 

 

 

 

Example

 

Rewrite log₄(ab)³

 

Use the exponents rule first.

3 log₄ab

 

Apply the rule of multiplication.

3(log₄a + log₄b)

 

Answer!

 

 

 

 

 

 

 

Example

 

Rewrite log₇xy² z⁴

 

Apply the rule of multiplication first to the x.

 

log₇x + log₇y² z⁴

 

Apply the rule of multiplication to the second term since the first term is already simplified.

 

log₇x + log₇y² + log₇ z⁴

 

Apply the rule of exponents to the second and third term to pull the exponents out front.

 

log₇x + 2log₇y + 4log₇ z

 

Answer!

 

 

 

 

 

 

Example

 

 

Rewrite the as an exponent.

log₆x^1/2

 

Apply the rule of exponents.

 

 

Simplify.

 

 

Answer!

 

 

 

 

 

 

 

 

 

 

 

Example

 

 

 

Apply the rule of exponents.

 

 

 

 

Apply the rule of division.

4(log₅3x - log₅y)

 

Apply the rule of multiplication to log₅3x.

4(log₅3 + log₅x - log₅y)

 

Answer!

 

 

 

 

 

 

 

 

Natural logs have the same properties of regular logs.

 

Multiplication

ln bc = ln b + ln c

 

Division

ln = ln b – ln c

 

Exponential

ln b^c = c ln b