Math Project.pdf - ~Log and Exponential Functions~ By Belen...

Unformatted text preview: ~Log and Exponential Functions~ By Belen Pair Period 6 Mrs. Taouk ~Graphing~ Graphing: Growth or Decay b>l = Growth 0<b<1 = Decay *b cannot equal 0 or 1 X- intercept Y- intercept More examples and help: Make y=0 Make X=0 Different Transformations Reflections If c is negative, reflect over the y-axis. If a is negative, reflect over the x-axis Shifts Vertical shift If k>0, the graph would be shifted upwards. If k<0, the graph would be shifted downwards. Horizontal Shift If h>0, the graph would be shifted left. If h<0, the graph would be shifted right. Up, down/left, right If h is positive go right If h is negative go left If y is positive go up If y is negative go down Stretch/ shrink If the absolute value of c is greater than one: horizontal shrink If the absolute value of c is less than one: horizontal stretch If the absolute value of a is greater than one: vertical stretch If the absolute value of a is less than one: vertical shrink ~Graphing Log functions~ y=a(logbc(x-h))+k y=a(lnc(x-h))+k domain → c(x-h)>0 Range → (-∞,+∞) VA → x=h Inverse Graphs Context The graph of inverse function of any function is the reflection of the graph of the function about the line y=xy=x . So, the graph of the logarithmic function y=log3(x)y=log3(x) which is the inverse of the function y=3xy=3x is the reflection of the above graph about the line y=xy=x . Example: If k=−3: The graph of y=[log2(x+1)] will be shifted 3 units down to get y=[log2(x+1)]−3. Extra Help: _52xEjT_k ~Graphing Exponential functions~ Exponential Functions y=a(b^c(x-h))+ k y=a(e^c(x-h))+ k Domain→(-∞,+∞) Range→y>k OR y<k HA→y=k Parent Function= b^x or e^x Y= 2^x Extra help: Example With Harder Transformation y= 2^(x+3) Replacing x with x+3 translates the graph 3 units to the left. 9DhlR43P7A iN1cGCgpdQ ~Expand and Condense log Expressions~ Natural Logs The natural logarithm of a number is its logarithm to the base of the e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parenthesis are sometimes added for clarity, giving ln(x), loge(x) or log(x). More Help: arithmic-functions/natural-logarithms-base-e/nat ural-logarithms/natural-log-definition Example: ~Expanding~ Log2[(8x^4)/5] → When asking you to expand an equation they want you to take apart 1 log equation into multiple. log2(8x^4) - log2(5) Why Did I Do This?: I split the numerator and denominator apart by converting the 1 log with division into Extra help with hard practice problems: es2.htm two logs using subtraction. Rules: ~Condensing~ Simplifying or combining logs The Product Rule: The logarithm of the product of numbers is the sum of logarithms of individual numbers. The Quotient Rule:The logarithm of the quotient of numbers is the difference of the logarithm of individual numbers. The Power Rule: The logarithm of an exponential number is the exponent times the logarithm of the base. The Zero Rule: The logarithm of 1 with b > 1 equals zero. The Identity Rule: The logarithm of a number that is equal to its base is just 1. Log of Exponent Rule: The logarithm of an exponential number where its base is the same as the base of the log equals the exponent. Exponent of Log Rule: Raising the logarithm of a number by its base equals the number Examples For The Most Used Rules Product Rule Examples This is the Product Rule in reverse because they are the sum of log expressions. We can convert those addition symbols into multiplication symbols inside the parentheses. Quotient Rule Examples The difference between logarithmic expressions tells us to use the Quotient Rule. I can put together x and 2 inside a single parenthesis using division operation. Harder Example: 1. Apply the Power Rule in reverse to condense the constants or numbers on the left of the logs. 2. Next step is to use the product and quotient rules from left to right. Work: Solving Exponential Equations and Logarithmic Equations One to One c/log1to1.htm Example: log​b​S=log​b​T if and only if S=T. How to Snail xponential-and.html Create a snail head around your log base b. And then you create the snail shell by raising this b to the power and being set equal to our last part. Solving Exponents Some exponential equations can be solved by rewriting each side of the equation using the same base. Other exponential equations can only be solved by using logarithms. MORE HELP: xA Example: solve this equation -> More Example Practice Problems 1) 3) 2) 4) Solving Logs Some logarithmic problems are solved by simply dropping the logarithms while others are solved by rewriting the logarithmic problem in exponential form. More Help: . htm Example: solve this equation -> More Example Practice Problems Solving Exponential and Log Functions Solving Logs are harder to teach. Using your knowledge from one to one functions and the snail method, try to practice solving. Another Example Quiz: Try it yourself Word Problems more help: Example of population growth Bacteria in a petri dish double in population every hour. Their initial population is 3. Assuming the absence of limiting factors, how many bacteria are there after 24 hours? Given: 3 bacteria in a petri dish They double every hour 24 hours Want: How many after 24 hours? Analysis: Let A=amount of bacteria after t hours A=3(2)t t=24 hours A=3(2)24 A=50331648 bacteria Example of compound interest Find the amount of money in an account with holding $200 at 5% interest compounded semiannually for 8 years. Given: Principle amount of money P = $200 Interest rate r = 0.05 Number of times compounded per year n = 2 Time t = 8 years Amount of money after t years A Want: A after 8 years Know: A=P (1+r / n)tn Analysis: A= 200(1+0.05 / 2)8*2 A=$296.90 Example of Half-Life Problems Isotope X has a half life of 300 years. After 900 years, how much of an original 200 gram sample is left? Given: 300 year half life 900 years Original 200g Want: How much is left after 900 years? Analysis: Let t = number of half lives Let A = amount of isotope after t years A=200(0.5t) t=900 years / 300 years = 3 half lives A=200(0.5)3 A=50 grams left The End ...
View Full Document