S56 (5.3) Logs and Exponentials.notebook March 02, 2016

Transcription

1 Daily Practice Today we will be learning about exponential and logarithmic functions. Homework due tomorrow. Need to know for Unit Test 2: Expressions and Functions Adding and subtracng logs, rewring logs without powers Solving equaons with logs Using addion formulae to find exact values Proving Trig. Idenes (2.) Wave funcon expressing in the form ksin(x ± a) or kcos(x ± a) Sketching graphs of transformaons (both degrees and radians) Wring down the funcon given a log graph Sketching trigonometric funcons with phase angle showing max, & min. and roots Wring down the equaon of a trig. Funcon given its graph (both degrees and radians) Composion of funcons h(x), g(x), f(x) etc. (2.2) Inverse of a funcon Expressing vectors in component form Proving Collinearity of vectors (2., 2.2) Dividing a line in a rao Finding the angle between two vectors Need to know for Unit Test 2: Expressions and Functions Adding and subtracng logs, rewring logs without powers Solving equaons with logs Using addion formulae to find exact values Proving Trig. Idenes (2.) Wave funcon expressing in the form ksin(x ± a) or kcos(x ± a) Sketching graphs of transformaons (both degrees and radians) Wring down the funcon given a log graph Sketching trigonometric funcons with phase angle showing max, & min. and roots Wring down the equaon of a trig. Funcon given its graph (both degrees and radians) Composion of funcons h(x), g(x), f(x) etc. (2.2) Inverse of a funcon Expressing vectors in component form Proving Collinearity of vectors (2., 2.2) Dividing a line in a rao Finding the angle between two vectors Logarithms Logs are the inverse function of exponential functions. y = f(x) y = x y = f - (x) Logarithms. Write in logarithmic form 25 /2 = 5 y = a b then b = log a y 2. p = a 3 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

2 Logarithms 3. Simplify log 4 64 y = a b then b = log a y 4. Change to exponential form a = log b q Ex. 5E Q 2 middle columns Q2,3 Daily Practice Today we will be learning about the Laws of Logs. Homework due! Laws of Logarithms These rules are not given in the exam. The logs must have the same base and be in the form log a x for the first 2 rules to work. Note also that Log a = 0 and Log a a =

3 Laws of Logarithms. Simplify log log log log log 2 3 Daily Practice Laws of Logarithms. 3. 2log 2 4-3log log 5 25 Logarithmic Equations Use the laws of Logs to help you solve these equations. Today we will be continuing to use laws of logs and solve equations with logs. Assembly 0.50 Assessment?. log a x - log a 5 = log a 20

4 Logarithmic Equations 2. log a x + 3log a 3 = log a 9 Daily Practice Today we will be completing to solve equations with Logs. Homework due Tuesday. Logarithmic Equations 3. Solve log 2 (x - 2) + log 2 x = 3 Logarithmic Equations 4. Solve 2log 9 x = / 2 + log 9 (5x + 8)

5 Daily Practice Graphs of Logarithmic Functions Remember: Logs are the inverse function of exponential functions. Today we will be learning about Logarithmic y = f(x) y = x Graphs. Assessment 7th March - Revision Material Online y = f - (x) Homework due Tuesday. Graphs of Logarithmic Functions Sketch the graph of y = log 3 x 5 4 x y Graphs of Logarithmic Functions Writing down the function given the graph. Use the graph shown to find the value of a when y = log a x 3 2 (3, ) (9, 2)

6 Graphs of Logarithmic Functions Writing down the function given the graph Q. DAILY PRACTICE Find the values of a and b for the function y = log a (x - b) (,.5) Transformation of log graphs Use techniques from functions and graphs topic.. Shown below is the graph of f(x) = log 3 x Today we will be learning how to transform graphs of logs. Homework due tomorrow. (i) State the value of a y (9, a) Assessment on Monday & Tuesday 7th, 8th March. 0 x (ii) Sketch the graph of f(x + 2) + Transformation of log graphs Daily Practice The graph shows the function f(x) = log Sketch the graph of g(x) = Sketch the graph of h(x) = log 5 5x, and show where the graph cuts the x-axis.

7 Functions that are of the form y = kx n will always have a curved graph. For many exponential graphs, graph becomes too large too quickly. Today we will be learning about graphs with logarithmic axes. Logs can be used to write these functions in the form y = mx + c and produce a linear graph. Homework Due today. Assessment Monday & Tuesday. The axes become logy and logx. Example: Express y in terms of x Any graph of 'logy' against 'logx' represents the equation y = kx n Given the function y = kx n, (where k & n are constants) take the log of both sides log 0 y (0, 5) gradient = 6 log 0 x Any graph of 'logy' against 'x' represents the equation y = ab x where a and b are constants. From the given graph, express y in terms of x. y = ab x gradient = 2.5

8 - y x S56 (5.3) Logs and Exponentials.notebook March 02, 206 Examples Example 2: Show that the data is related by the formula y = kx n, then find the values of k and n x y log 0x log 0y y x Daily Practice Experimental data are given in the table below: Show that the formula connecting y and x is of the form y = ab Find the value of 'a' and 'b' and state the formula that connects x and y. (ie Find a formula for y in terms of x) Solution: (a) x log 0y Creates a linear equation => related by the formula y = ab x y = 2 x 0.8 x (b) - y = ab x log 0y = log 0a + xlog 0b y = c + mx = c + 0.5m 0.85 = c +.2m = -0.7m m = log 0b = b = = c + 0.5(-0.097) = c c = log 0a = c a = = 2 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 Today we will be learning about logs to the base e. Assessment next Monday & Tuesday. e is a value for which given the function y = e x, the derivative of the function is the exact same. e is also used in formulae for compound interest with continuous compounding. It looks like e x on your calculator or sometimes exp(x)

9 Solving Equations with e Natural Logs Logs that have base e are called natural logs that can be written as log e x or ln x. Find ln 8 to 3 decimal places To solve equations with e: Simplify the equation if possible log e e = Take the natural log of both sides Use the rules of logs.. Solve e x = e.2t = Solve ln x = 2 3. Solve ln x = 0.84 Questions in Context It is claimed that a wheel is made from wood which is over 000 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. 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? mark For the wheel it was found that A(t) was 88% of the amount of carbon in a living tree. (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 Is the claim true? 5 marks Questions in Context Mixed Questions on Logs

10 204 Specimen paper