Introduction to Logs

Logarithms

What are they and how do you work with them?

Logarithms are Exponents

• log is short for logarithm

• when you perform a log function on a number the result is an exponent

• a log has a base to it just like the base of an exponent

• performing a log function is the inverse of an exponential function

Base of a Log

• The base of a log is the same as the base of an exponential

• log₂ 8= y means the same as 2^y=8

• We would read the above as log base 2 of 8

• Notice that 2 to the 3rd power is 8, so y=3 in the above equations

• so log₂8=3

• log₅25 = 2 since 5²=25

The Common log

• if we write a log without a base such as log1000=3, this means the base is 10

• Since using a base of 10 is very common we call this the common log.

• log1000 means the same as log₁₀1000

• Our number system is base 10

• 10⁰=1 10¹=10 10²=100 10³=1000

• So when we write log x it automatically means we are dealing with base 10

Log form vs Exponential form

• Watching the gif....

• Notice how base 2 stays the base of the exponential equation

• Notice how the result of the log, the ? becomes the exponent

• Notice how the 64, which we are taking the log of, becomes the result of the exponential

Logs are inverse functions of Exponents

• Just like subtraction is inverse to addition

• Division is inverse to Multiplication

• Square root is inverse to squaring

• Applying a log with the same base as an exponential is the inverse to using the exponent and can tell us what the exponent is.

• For example log250 = x tells us what the exponent of 10^x = 250 would be. Plug this into your calculator to find the exponent.

Multiple Choice

Change the following exponential equation into a log equation:

$$2^5=32$$  

• $$\log_25=32$$  

• $$\log_{32}5=2$$  

• $$\log_232=5$$  

• I hate logs. Nobody likes them.

Multiple Choice

Change this log equation into an exponential equation:

$$3^4=81$$  

• $$\log_34=81$$  

• $$\log_381=4$$  

• $$\log_{81}3=4$$  

• 3 frogs sat on 4 logs and floated away from 81 dogs

Multiple Choice

find the answer to the following expression:

$$\log_5625$$  

• 125

• 5

• 7

• 4