7.2 Notes: Log basics

Exponential Functions: Exponential functions have the variable

located in the exponent spot of an equation/function.

EX: 2x = 6 32x-7 = 98 72x = 54

So, what is a logarithm? Well, if we were given 2x = 4, we could figure

out that x is 2. If we were given 3x = 27, we could figure out that x = 3. But what about 2x = 6? Do we know what power 2 is raised to to make 6?

How do we solve this then? Well, just like we would solve any other equation (3x + 7 = 19), we use OPPOSITE OPERATIONS.

The opposite of an exponent is a logarithm

Logarithmic form: The log form is: logby = x

Translating between forms: Exponential form: Logarithmic

form: bx = y logby = x

“b” is the base “x” is the exponent “y” is the “answer”

Examples: Change into log form: A) 3x = 9 B) 7x = 343 C) 5x =

625

Change into exponential form: D) log6a = 2 E) log416 = y F)

log327 = t

Common and Natural Logs The only difference between common logs and

natural logs is the base. The common log has a base of 10. Just like ones,

the base of 10 is not written and understood. Log10x = log x

The natural log has a base of “e.” It is not written and understood to be the base.

Logex = ln x

Can we find these answers in the calculator? ABSOLUTELY! The calculator recognizes only

base 10 and base e logarithms. Let’s find the buttons…..

EX: log 8 ln 0.3 log 15 ln 5.72

What do these mean? What are they asking?

Log Properties…… Just like algebra has properties (commutative,

associative, identity, etc….), logarithms have properties as well. They help us solve equations involving logarithms.

Product Rule: logbmn = logbm + logbn

EX: log 7x (what’s the base??) =

EX: log23t =

The Quotient Rule: = logbm – logbn

EX: =

EX: =

n

mblog

3log7

m

25log5

c

The Power (Exponent) Rule:

Logbmn = n ∙ logbm

EX: log3r5 =

EX: log4v2/3 =

Inverse properties: Inverse properties are opposites, they “un-do”

each other’s operation.

A) logbbx = x B) = x

EX: log774 = =

EX: log11116 = =

xbblog

3log99

4

1log88