What is the relationship between powers, roots and logarithms?

How do you find an inverse of an equation?

Find the inverse of y = 2x + 3 x = 2y + 3 x – 3 = 2y (x-3) = y -1 1) Switch the x and y 2) Solve for the new y also known as f -1 (x) 2

 x 0 = 1 x≠ 0  x -1 =  x n ∙x m = x n+m  = x n-m  (x n ) m = x nm   when taking the root of a variable you can’t take the even root of a neg #

x 2 = 49

rationalizesolve

 Worksheet 1

What is the relationship between logarithms and exponential functions?

What do we do with problems like the last one on the homework y = 3 x x = 3 y log x = log 3 y Not as easy to solve for y when y is the exponent so we remember the primary rule of equations: whatever we do to one side we must do to the other. In this case we take the logarithm of both sides

log b x = y becomes x = b y Solve: log 2 4 = x log 2 x 3 = 3 log 1000 = x

log = x log 4 = x log.1 = x

 Worksheet 2

How are the laws of logarithms related to the properties of exponents?

Primary rule of logs: log b x = y becomes x = b y What would be true of the following and WHY???? log a x = 0 log a a = 1 NOTE: Can’t take the log of a negative number i.e. in log b x = y the x can’t be negative why?

let b = log a x and c = log a y convert x=a b y = a c multiply xy =a b a c xy = a b+c log a xy =log a a b+c convert log a xy = b + c substitute log a xy = log a x + log a y

log (x 2 -1) – log (x+2) = 1 log (4x -4) log x =2

 Worksheet 3

 What is the difference between these three problems and how does that impact the way you work with them? log 50 + log 2log x = log 12 – log 3log 8 – log x = 2

 Worksheet 4

How can logarithms assist in solving an exponential equation?

What is THE primary rule of equations —whatever you do to one side you must do to the other. 3=4 x log 3=log 4 x ln 3=ln 4 x

After using the circular method, you see are you back to solving exponential equations. 3=4 x log 3=log 4 x log 4 3=x log 3=x log 4 log 3=x log 4

 Worksheet 5

What are the real world applications of exponential and logarithmic equations?

 I = Prt   A= final amountI = interest  P = principalP = principal  r = rate as a decimalr = rate as a decimal  n = number of times compounded in one year t = time in years  t = the time in years   How are they the same and how are they different:

 In 1900, the population of the U.S. was 3,465,000 with an annual growth rate of 6.2%. How long will it be until the population reaches 10,000,000?

 A certain bacteria colony has a growth rate of 26% per hour. If there were 42 bacteria in the colony when the study began, how long will it take to have 258 bacteria?

 In 2000, the population of a county in Southeastern PA was 5,263,126. The population of this area has been decreasing at a rate of 3% per year, if this continues, when will the population go below 4,500,000?

 Worksheet 6

 Worksheet 7