Exercise 21: Logarithmic Functions Lessons

What is Logarithmic Functions?

Logarithmic Functions are the inverse of exponential functions

(y = b ^x)

Recall: In order to find the inverse of a function, we follow this equation, y = x. Which means that all y – values are x – values and all x – values are y – values.

So by knowing this, the inverse of y = b ^x should be x = b ^y, however, x = b ^y creates a problem though because “y” need to be isolated but “y” is in the exponent position.

So by using Logarithmic Functions, x = b ^y is express as y = log _b x

Summary:

Logarithmic Functions of y = log _b x is the inverse of y = b ^x where b cannot equal to 1, b > 0 and x > 0

Step by process: y = b ^x >> x = b ^y >> y = log _b x

y = Logarithm (Exponent)

b = Base

x = Argument (Power)

Note: The “log” part of functions are not variables, they only tells us that we are dealing with Logarithmic Functions.

An example of exponential statement into logarithmic form

36 = 6²

Since, 36 is the Power

6 is the Base

2 is the Exponent

Then the answer should be…

2 = log ₆ 36

An example of logarithmic statement into exponential form

Log ₅ 625 = 4

Since, 5 is the Base

625 is the Power

4 is the Exponent

Then the answer should be…

5 ⁴ = 625

These next few examples are design to trick you

Example 1:

y = 4 ^x

If you look closely, we see that the x and y changes place according to the summary. It seem like this question is changing the rule but this does not change what is the “Exponent” and what is the “Power” is.

Since y is the Power

4 is the Base

x is the Exponent

Then the answer should be…

x = log ₄ y

Example 2: log ₂ (-16) = x

There is a problem with the -16 as a power. It is impossible to get -16 because the argument (Power) cannot be zero or lower (negative). So we simple state that there is “no solution”.

Prove: log ₂ (-16) = x

2 ^x = -16

No Solution

Example 3: log ₅ 1/125

The 1/125 is just the inverse of 125. So 125 of a base of 5 is equal to 5 ³, therefore, 1/125 is 5 ^{– 3}

So, the answer is…

log ₅ 1/125 = 5 ^{– 3}

How to do Logarithmic Functions on a calculator:

The log button on all calculators is a base of 10 or in other words: log₁₀

If we want to find the exponent of y = log _b x, we need to cancel out the base 10

Example:

y = log ₂ 8

y = ( log ₁₀ 8 ) / ( log ₁₀ 2)

y = ( log 8) / ( log 2 )

y = 3

Graphing Logarithmic Functions:

When graphing Logarithmic Functions, we take the inverse of y = b ^x and then shift the graphs normally as a transformation question.

Example 1:

y = 2 ^x = log _{2 (x)}

y = 2x
x	y
-2	¼
-1	½
0	1
1	2
2	4

y = log 2 x
x	y
¼	-2
½	-1
1	0
2	1
4	2

Example 2:

y = 2 ^x = log ₂ (x – 1)

Remember that the signs inside of the bracket shifts the graphs horizontally but opposite. So all x values shift to the left by 1.

Y = log 2 (x – 1)
X	y
5/4	-2
3/2	-1
2	0
3	1
5	2

Example 3:

y = 2 ^x = log ₂ (x) + 2

The graph will be moving 2 units up.

Y = log 2 x
x	y
¼	0
½	1
1	2
2	3
4	4