Logarithms are Exponents

• log is short form of 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

Logarithms to base a

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.

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

Rules

Examples

Examples