Warm Up Graph the following 1 fx 2

Warm Up: Graph the following: 1) f(x) = 2 e. 4 x 2) y = e-1. 4 x

6. 3 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?

HW: p. 314 #5 – 16, 27 - 32

Warm Up:

Inverse properties: Inverse properties are opposites, they “un-do” each other’s operation. A) logbbx = x B) =x EX: log 774 = = EX: log 11116 = = EX: log 525 x =

Finding Inverse Functions: Remember, when we found inverse functions before break, we did the following steps: A) Swap the x and y B) solve for y using inverse (opposite) operations C) Simplify the answer if necessary EX: 1) f(x) = 6 x 3) h(x) = ex 2) y = ln(x + 3) 4) y = log(x + 6)

Exponential Graph Transformations: Describe (in words) the transformation(s), sketch the graph and give the domain and range: 1) g(x) = ex+4 + 2 2) y = -(½)x - 3

Graphing Log Functions (by hand)

Log Graph Basics: Because the equations of logarithms are inverses (opposites) of exponential equations, the basics of the graphs are also inverses (opposites). “Go – to” point is (1, 0) Vertical asymptote at x = 0 To graph by hand, rewrite the log into an exponential equation, make a table of values, then use the inverse of the table (swap the x and y values) to graph the log function.

Examples: (don’t forget to give the D & R!) 1) f(x) = log 3 x 2) g(x) = log½x

Graph: h(x) = log 5 x a(x) = log ¾ x

p. 315 #35 -42, 55 -59

Warm Up: Rewrite