Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Logarithms are useful in order to solve equations in which the unknown appears in the exponent Exponent is another word for index. The variable x is the index (exponent) Exponent is the logarithm Inverse Base is always the base

Reflection points, off y=x Exponent is the logarithm. The output/y-value Corresponding input/x-value Input 4 produces a smaller output 2  a log function grows slower than an exponential function

Objectives • Understand the idea of continuous exponential growth and decay • Know the principal features of exponential functions and their graphs • Know the definition and properties of logarithmic functions • Be able to switch between the exponential and logarithmic forms of an equation • Understand the idea and possible uses of a logarithmic scale • Be familiar with the logarithms to the special base e and 10 • Be able to solve equations and inequalities with the unknown in the index • Be ale to use logarithms to identify models of the form y = abx and y = axn.

Logarithms Common (or Briggian) logarithm (log) of the number. Base 10 logarithm Natural (or Napierian) logarithms How many fingers do you have? What is the base of our number system? Our number system is based on powers of 10 Logarithmic tables have values between 1 and 10 3 is the common logarithm of 1000, since 103 = 1000. The base is 10. Exponent is the logarithm. Adding 2+1 is easier than multiplying 100 x 100  this is why the logarithms were invented in the 17th century  multiplication by addition and division by subtraction. A Table of the Common Logarithm http://www.sosmath.com/tables/logtable/logtable.html

Logarithmic Scale Amount of cars on I95 Percentage wise, 100 to 10,000 is much larger increase than 50,000 to 1000,000 Logarithmic scale is suitable for large data swings, such as here, the number of cars on the highway goes from 100 to 1,000,000 The highest 9900% increase in traffic is not clear at all on a linear scale. One would wrongly conclude that the largest day-to-day percent increase happened on Friday The biggest difference in the daily graph heights occurred on Monday-to-Tuesday, not on Thursday-to-Friday. Amount of cars on I95 – Linear Scale Amount of cars on I95 – Logarithmic Scale All data is clearly seen Hard to see anything here

Continuous Exponential Growth/Decay Gets small in a hurry as x gets bigger. ½ When x is 1; 1024 when x is 10 number of time unites after start Initial value x = (0:.1:10) y = 10.*(.5.^x) plot (x,y) Graph never touches the x axis Initial value; Anything to the power 0 is a one rate of growth (r>1) rate of decay (r<1) Exponential Decay: radioactivity in lump of uranium ore, concentration of an antibiotic in the blood stream discrete growth continuous growth A geometric sequence with common ratio r. Functions having natural numbers. Functions having real numbers. • Exponential Growth: • Rampant inflation • A nuclear chain reaction • Spread of an epidemic • Growth of cells x = (0:.1:4) y = 10.*(2.^x) plot (x,y) Exponential Growth

Example U.S. population in 1790: 3.9 million (initial value) U.S. population in 1990: ??? million Number of years Population  b = 1.030… U.S. population in 1860: 31.4 million We could avoid solving for b

What percent of carbon-14 did we loose in 100 years – given it’s half life Decay of isotope carbon-14’s half life: 5715 years By what percentage does carbon-14 decay in 100 years? 0.5 units are left after 5715 years x units are left after 100 years We lost .012 units .012/1 = 0.012 = 1.2%

Properties of Logarithms exponent is the logarithm Lookup log10 2 in the Table of the Common Logarithm. http://www.sosmath.com/tables/logtable/logtable.html Example: same

Example: Logarithmic Functions How to find the logarithm of 3456 with a log table with values between 1 and 10 Please note this In the old days, without calculators how could one find the cube root of 100 ? All they did was one lookup and a simple division log of 4.64 is 0.6665180 reverse lookup (table of inverse function) do the log first and then undo the log Inverse exponent is the logarithm

logb exponential form  log10 Base 2 This is straightforward Not straightforward Convert to exponential form Take the logarithm on both sides Base 10 Power rule

Inequalities t days How many days does it take for the amount to fall less than 0.1 units? Take the logarithm on both sides a negative number switch the direction of inequality The iodine-131 will fall to less than 0.1 units after about 26.7 days.

Example 2, 3 – page 299 -- ax and a-x Graphs reflect on the y axis decays faster Grows faster Becomes clear as to why negative exponent is a decreasing function All intercept here (0,1) Anything to the power 0 is a one x axis is the horizontal asymptote

Example 4 – page 301 -- Transformations Shift f(x) one unit to the left: add one to the input. Input is a bigger number  takes off faster Shift f(x) down by 2 units: subtract 2 from the input reflect f(x) on x axis: take the output and multiply it by -1 reflect f(x) on y axis: input is multiplied by -1 (negative input)

Graphs of Exponential Growth Output is growing exponentially a, b are constants Representing an exponential function as a linear function Taking the logarithm of both sides, to any base slope intercept

Amount invested: $1000 Annual interest: 6% One year interest = 1000 x .06 = $60 Amount at the end of the year = $1000 + $ 60 = $1060 = 1000 x 1.06 Geometric Series number of compounds per year Number of years For large n; continuous compounding

Example 6 (p314) Transformation of Graphs of Logarithmic Functions Add two to the output Shift f(x) one unit to the right

exponent is the logarithm Properties changing base logarithm Log functions and exponential functions are inverse of each other transformations

Exponent is the logarithm base is same uncommon base Exponent is the logarithm Bases are same

Excel has built-in functions to calculate the logarithm of a number with a specified base, the logarithm with base 10, and the natural logarithm.

Exponential and Logarithmic Functions