Log Change of Base

Learning Objective

• understand the relationship between logarithms and indices.

• able to use the laws of logarithms to any base (excluding change of base)

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

Multiple Choice

Write the following equation in logarithmic form:

3⁵ = 243

• log₃5 = 243

• log₂₄₃3 = 5

• log₅243 = 3

• log₃243 = 5

Logarithms to base a

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.

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

Multiple Choice

Rewrite the equation in exponential form:

log₅125 = 3

• 125³ = 5

• 5¹²⁵ = 3

• 5³ = 125

• 3⁵ = 125

Multiple Choice

Solve the equation for x.

• x = 27

• x = 9

• x = 3

• x = 1

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

Evaluate log₅25 = x

• x = 2

• x = 5

• x = 125

Multiple Choice

Evaluate: log₉3 = x

• ½ = x

• -½ = x

• 2 = x

• -2 = x

Multiple Choice

Change this an exponential equation into log 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

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

Multiple Choice

log √1000 =

• 1/2

• 3/2

• 5/2

• 2

Multiple Choice

log 10^(1/2) =

• 1/2

• 1/3

• 1

• 2

Examples