CALCULUS CHAPTER 1 PT 2 Unit Question: What is a logarithmic function and how are they used to solve real-world problems?

SECTION 1-5 DAY 1 EQ: What is the relationship between exponential equations and logarithmic equations?

FUNCTIONS AND LOGARITHMS

EXAMPLE Given f(x) = a x find f -1 (x) y = a x x = a y How can this be solved?

DETERMINING IF 2 EQUATIONS ARE INVERSES

SYMMETRY TO Y=X  If f -1 (x) = f(x) then there is symmetry to the line y = x EX: 3x + 3y = 5equation4x – 4y = 8 3y + 3x = 5inverses4y – 4x = 8 Is there symmetry to y = x?

HW: Worksheet 1-5-1

SECTION 1-5 DAY 2 EQ: What are the Properties of Logarithms? EQ: How do you solve equations involving logarithms? EQ: What is a natural log?

FINDING THE INVERSE OF AN EXPONENTIAL 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

PROPERTIES OF LOGARITHMS Primary Rule of Logarithms log b x = y becomes x = b y Solve: log 2 4 = x log 2 x 3 = 3 log 1000 = x

USING THE PRIMARY RULE: What would be true of the following and WHY???? log a x = 0 log a a = x means x =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?

PROVING RULES OF LOGARITHMS 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

ADDITIONAL RULES

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

SOLVE USING THE RULES OF LOGARITHMS log 50 + log 2=xlog x = log 12 – log 3log 8 – log x = 2

CHANGE OF BASE FORMULA

SOLVE: LOGARITHMS AND NATURAL LOGS 3=4 x log 3=log 4 x log 3 = x log 4 ln 3=ln 4 x ln 3 = x ln 4

FINAL RULES

HW: Worksheet 1-5-2

SECTION 1-5 DAY 3 EQ: What are real-world applications of exponential and logarithmic functions?

OTHER FORMULAS  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:

EXAMPLES  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?

EXAMPLES  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?

HEADING  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?

EXAMPLES Knowing P = P 0 e rt The approximate population of Dallas was 680,000 in In 1980 it was 905,000. Find r in the growth formula and use it to approximate the population in 2010.

HW: Pg to 49

SECTION 1-4 DAY 1 EQ: What are parametric equations and how are they used?

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HW: Pg. 39