7.2 Notes: Log basics

Exponential Functions:  Exponential functions have the variable located in the exponent spot of an equation/function.  EX: 2 x = 632 x-7 = 987 2x = 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 (3x + 7 = 19), we use OPPOSITE OPERATIONS.  The opposite of an exponent is a logarithm

Logarithmic form:  The log form is: log b y = x  Translating between forms:  Exponential form:Logarithmic form:  b x = ylog b y = x  “b” is the base  “x” is the exponent  “y” is the “answer”

Examples:  Change into log form:  A) 3 x = 9B) 7 x = 343C) 5 x = 625  Change into exponential form:  D) log 6 a = 2E) log 4 16 = yF) log 3 27 = 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.  Log e x = 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 8ln 0.3log 15ln 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: log b mn = log b m + log b n  EX: log 7x (what’s the base??) =  EX: log 2 3t =

 The Quotient Rule: = log b m – log b n  EX: =

 The Power (Exponent) Rule:  Log b m n = n ∙ log b m  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) log b b x = xB) = x  EX: log = =  EX: log = =