7 2 Notes Log basics Exponential Functions Exponential

7. 2 Notes: Log basics

Exponential Functions: Exponential functions have the variable located in the exponent spot of an equation/function. EX: 2 x = 6 32 x-7 = 98 72 x = 54

So, what is a logarithm? Well, if we were given 2 x = 4, we could figure out that x is 2. If we were given 3 x = 27, we could figure out that x = 3. But what about 2 x = 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 (3 x + 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) 3 x = 9 B) 7 x = 343 C) 5 x = 625 Change into exponential form: D) log 6 a = 2 E) log 416 = y F) log 327 = 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. Log 10 x = 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 7 x (what’s the base? ? ) = EX: log 23 t =

The Quotient Rule: EX: = = logbm – logbn

The Power (Exponent) Rule: Logbmn = n ∙ logbm EX: log 3 r 5 = EX: log 4 v 2/3 =

Inverse properties: Inverse properties are opposites, they “un-do” each other’s operation. A) logbbx = x B) =x EX: log 774 = = EX: log 11116 = =