S56 (5.1) Logs and Exponentials.notebook October 14, 2016

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1 1. Daily Practice Exponential Functions Today we will be learning about exponential functions. A function of the form y = a x is called an exponential function with the base 'a' where a 0. y = a x base exponent (index) y x Because the x - axis represents the power, these graphs become very steep very quickly. Exponential Functions If a > 1, y = a x is known as a growth function. If 0 < a < 1, y = a x is known as a decay function. 1 Exponential Functions The equation of an exponential function can be found using points from its graph. 1. State the equation of the function shown if it's in the form y = a x The point (0, 1) will be on every exponential graph of the from y = a x. 1

2 Exponential Functions 2. State the equation of the function shown if it's in the form y = a x + b Exponential Functions 3. Draw a sketch of the graph y = 3 x + 1 Ex. 3N 1. Daily Practice Today we will be learning about logarithmic functions. 2. Logarithms Logs are the inverse function of exponential functions. y = f(x) y = x 1 1 y = f -1 (x) If y = a b then b = log a y where b = log a y is called the logarithmic function of b to the base a E.g. if 8 = 2 3, then 3 = log 2 8

3 Logarithms Write in logarithmic form /2 = 5 y = a b then b = log a y 2. p = a 3 Logarithms 3. Simplify log 4 64 y = a b then b = log a y Daily Practice Q1. Sketch the graph of y = 2cos(x - ) for 0 x 2π Q2. 4. Change to exponential form a = log b q Today we will be learning about the Laws of Logs.

4 Laws of Logarithms Laws of Logarithms: Proof of Multiplication Rule Log a xy = log a x + log a y Log a = log a x - log a y Log a x n = nlog a x These rules are not given in the exam. The logs must have the same base and be in the form 1log a x for the first 2 rules to work. Note also that Log a 1 = 0 and Log a a = 1 Laws of Logarithms: Proof of Multiplication Rule Laws of Logarithms 1. Simplify log log 3 9 Log a xy = log a x + log a y Log a = log a x - log a y Log a x n = nlog a x 2. log log log 2 3 Laws of Logarithms 3. 2log 2 4-3log 2 2 Log a xy = log a x + log a y Log a = log a x - log a y Log a x n = nlog a x Daily Practice Write the function y = 3x x - 5 in the form a(x + p) 2 + q 2. State the equation of the line that passes through (1, -4) and is parallel to 3x - 2y = 7 3. Simplify log a p + log a q 4. 2log 5 25

5 Today we will be continuing to use laws of logs and solve equations with logs. Logarithmic Equations Use the laws of Logs to help you solve these equations. Log a xy = log a x + log a y Log a = log a x - log a y 1. log a x - log a 5 = log a 20 Log a x n = nlog a x Logarithmic Equations 2. log a x + 3log a 3 = log a 9 Log a xy = log a x + log a y Log a = log a x - log a y Log a x n = nlog a x Logarithmic Equations Logarithmic Equations 3. Solve log 2 (x - 2) + log 2 x = 3 where x > 0 Log a xy = log a x + log a y Log a = log a x - log a y 4. Solve 2log 9 x = 1 / 2 + log 9 (5x + 18) where x > 0 Log a x n = nlog a x

6 If Log a x = log a y then x = y Daily Practice Today we will be continuing to solve equations with Logs. Next Unit Assessment?? DAILY PRACTICE Q1.

7 Graphs of Logarithmic Functions Sketch the graph of y = log 3 x 5 x y Today we will be learning about Logarithmic Graphs Graphs of Logarithmic Functions Writing down the function given the graph 1. Use the graph shown to find the value of a when y = log a x Graphs of Logarithmic Functions Writing down the function given the graph 2. Find the values of a and b for the function y = log a (x - b) (11, 1.5) Transformation of log graphs Use techniques from functions and graphs topic. 3. Shown below is the graph of f(x) = log 3 x y (i) State the value of a Transformation of log graphs 4. The graph shows the function f(x) = log Sketch the graph of g(x) = (9, a) Sketch the graph of (ii) Sketch the graph of f(x + 2) x h(x) = log 5 5x, and show where the graph cuts the x-axis.

8 Daily Practice Q1. Write 7x x - 8 in completed square from Domain: These are all the possible values of x. We need to choose a domain that makes the expression/equation possible or valid. Q2. Show that x - 1 is a factor of x 3-3x and hence solve x 3-3x = 0 Range: This is the range of values for y or the output when x has been substituted in. Q3. Find the points where the line y = 4x + 8 and the curve y = x 3-5x + 8 intersect You may need to think about what is suitable for the domain or range. Try to think about what values of x or y aren't possible or don't work. E.g A suitable domain would be {x E R, where x 2} or for y = (x - 5) 2 + 2, a suitable range would be y 2

9 Daily Practice Today we will be learning about graphs with logarithmic axes. Some Log logic... Q1. Daily Practice Remember how we discussed only using positive bases with exponential functions because they can't be graphed otherwise. b = a y Logs are the inverse of exponential functions which means that their base must always be positive. This means that Q2. for y = log a b a > 0 and b > 0 always! Graphing with Logarithmic axes Logarithmic scales and axes make it easier to compare data over a very large range. They are used in many real world examples such as earthquake magnitude, medicine concentration and the finance industry. Today we will be learning about graphs with logarithmic axes. Logs can be used to write functions such as y = ax b and y = ab x in the form y = mx + c and produce a linear graph.

10 Graphing with Logarithmic axes When the axes are 'logy' and 'logx', this represents the equation y = ax b Graphing with Logarithmic axes Example: Express y in terms of x Given the function y = ax b, (where a & b are constants) we take the log of both sides log 10 y gradient = 6 (0, 5) log 10 x Daily Practice Showing data is related 1. If the graph of log y against log x is a straight line, then type y = ax 2. If y = mx + c can be written in terms of log y and log x (ie a linear equation) then this confirms that the formula connecting y and x is of the type y = ax Today we will be continuing to learn about related functions. Assessment: Friday 14th October?

11 Graphing with Logarithmic axes Example: Show that the data is related by the formula y = kx n, then find the values of k and n x y Daily Practice Creates a straight line graph, this means they are related by the function y = kx n Today we will be continuing to learn about graphs with Logarithmic Axes. Graphing with Logarithmic axes Any graph of 'logy' against 'x' represents the equation y = ab x where a and b are constants. Graphing with Logarithmic axes From the given graph, express y in terms of x. gradient = 2.5 y = ab x

12 Daily Practice Q1. The function f(x) = x is shown in the diagram (i) State f -1 (x) (ii) Draw the graph of f -1 (x) Q2. The graph opposite is of Today we will be continuing to practise questions on log graphs. the form y = kx n, state the values of k and n Graphing with Logarithmic axes Example: Experimental data are given in the table below: Show that the formula connecting y and x is of the form y = ab and hence find the values of a and b Daily Practice Q1. Below is a drawing of the graph y = acos(x + b), state the values of a and b Today we will be learning about logs to the base e. Q2. Q3.

13 The Exponential Function The exponential function most often refers to the 'natural' exponential function y = e x where e is a constant (like π) known as Euler's number and whose value is approximately It appears in many parts of Maths including compound interest & growth and decay functions. Natural Logs Logs that have base e are called natural logs that can be written as log e x or ln x. We can use them to work out problems on growth & decay. 1. Find ln 8 to 3 decimal places It looks like e x on your calculator or sometimes exp(x) 2. Solve ln x = Solve ln x = 0.84 Solving Equations with e To solve equations with e: Simplify the equation if possible log e e = 1 Take the natural log of both sides Use the rules of logs. 1. Solve e x = e 1.2t = 74.9 Questions in Context It is claimed that a wheel is made from wood which is over 1000 years old. To test this claim, carbon dating is used. The formula A(t) = A 0 e t is used to determine the age of the wood, where A 0 is the amount of carbon in any living tree, A(t) is the amount of carbon in the wood being dated and t is the age of the wood in years. For the wheel it was found that A(t) was 88% of the amount of carbon in a living tree. Is the claim true? 5 marks 1. Daily Practice

14 The value, V (in million) of a cruise ship t years after launh is given by the formula V = 252e t (a) What was the value when launched? 1 mark Today we will be finishing logs. (b) The owners decide to sell the ship once its value falls below 20 million. After how many years will it be sold? 4 marks 1. Daily Practice Pg. 289 or 299 Q4-7 Specimen paper Higher Unit Assessment Topics Functions: Composite functions, inverse functions, Today we will be starting revision for the assessment. Graphs transformations of functions including log and trig. graphs Logs: Laws of logs, simplifying logs, writing the equation from a graph.

15 Daily Practice Q1. The graph illustrates the law y = kx n. If the straight line passes through A(0.5,0) and B(0,1), find the values of k and n Daily Practice Q1. Simplify 5log a p - 4log a p a,p > 0 Q2. Given f(x) = 5x 2-2, find f -1 (x) the inverse of f(x) Q2. Simplify 3log e 2e 2log e 3e expressing your answer in the form A + log e B log e C, where A, B and C are whole numbers Q3. Draw the graph of y = 2sin(x - ), where 0 x 2π