Logs and Exponentials Higher.notebook February 26, Daily Practice

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1 Daily Practice Daily Practice

2 Today we will be learning about exponential functions and logs. Homework due! 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 (2.1) 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 (2.1, 2.2) Dividing a line in a rao Finding the angle between two vectors

3 Exponential Functions A function of the form y = a b is called an exponential function to the base 'a' where a 0. 'a' is known as the base and 'b' is known as the exponent or index. If a > 1, y = a b is known as a growth function. 1 If a < 1, y = a b is known as a decay function. 1 Exponential growth implies that there is a consistent fixed period over which the function will change by a fixed proportion. E.g. 4 1 = 4, 4 2 = 16, 4 3 = The function quadruples in size every time you increase the power by 1 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

4 Logarithms Examples: y = a b then b = log a y 1. Write in logarithmic form 25 1/2 = 5 2. p = a 3 Logarithms 3. Simplify log 4 64 y = a b then b = log a y 4. Change to exponential form a = log b q

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6 Today we will be learning about the Laws of Logs. Homework Online due Laws of Logarithms Log a xy = log a x + log a y Log a = log a x - log a y These rules are not given in the exam. Log a x n = nlog a x 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

7 Laws of Logarithms Examples: 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 Examples: 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 4. 5log 5 25

8 Logarithmic Equations Use the laws of Logs to help you solve these equations. Examples: 1. log a x - log a 5 = log a 20 Log a xy = log a x + log a y Log a = log a x - log a y Log a x n = nlog a x

9 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 3. Solve log 2 (x - 2) + log 2 x = 3 Log a xy = log a x + log a y Log a = log a x - log a y Log a x n = nlog a x

10 Logarithmic Equations 4. Solve 2log 9 x = 1 / 2 + log 9 (5x + 18) Daily Practice

11 Today we will be learning about Logarithmic Graphs. Graphs of Logarithmic Functions Remember: Logs are the inverse function of exponential functions. y = f(x) y = x 1 1 y = f -1 (x)

12 Graphs of Logarithmic Functions Sketch the graph of y = log 3 x 5 x y Graphs of Logarithmic Functions Writing down the function given the graph Examples: 1. Use the graph shown to find the value of a when y = log a x (3, 1) (9, 2)

13 Graphs of Logarithmic Functions Writing down the function given the graph Examples: 2. Find the values of a and b for the function y = log a (x - b) (11, 1.5)

14 Daily Practice Today we will be learning about logs to the base e. Homework Online due

15 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 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) Natural Logs Logs that have base e are called natural logs that can be written as log e x or ln x Examples: 1. Find ln 8 to 3 decimal places 2. Solve ln x = Solve ln x = 0.84

16 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. Examples: 1. Solve e x = e 1.2t = 74.9 Questions in Context

17 Q1. DAILY PRACTICE Q2. Today we will be learning about graphs with logarithmic axes.

18 Graphing with Logarithmic axes 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 so instead, Logs can be used to write these functions in the form y = mx + c and produce a linear graph. The axes become logy and logx. Given the function y = kx n, (where k & n are constants) take the log of both sides Graphing with Logarithmic axes Examples

19 Graphing with Logarithmic axes 2. Express y in terms of x log 10 y gradient = 6 (0, 5) log 10 x Daily Practice

20 Today we will be continuing to learn about graphing with logarithmic axes. Homework Online due Graphing with Logarithmic axes 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 10 x log 10 y y x

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22 Graphing with Logarithmic axes If data is related in the form y = ab x where a and b are the constants. We can also rewrite this using logs so that it is in the form of a linear equation. (In this case, when making a new table, you only need to find log 10 y and keep x as it is) We can then find the values of a and b. y = ab x Graphing with Logarithmic axes Examples: 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 10 y y 1 x Creates a linear equation => related by the formula y = ab x y = 2 x 0.8 x (b) - y = ab x log 10 y = log 10 a + xlog 10 b y = c + mx = c + 0.5m = c + 1.2m = -0.7m m = log 10 b = b = = c + 0.5( ) = c c = log 10 a = c a = = 2

23 Graphing with Logarithmic axes Any graph of 'log y' against 'x' represents the equation gradient = 2.5 From the given graph, express y in terms of x. Transformation of log graphs Use techniques from functions and graphs topic. Examples: 1. Shown below is the graph of f(x) = log 3 x y (i) State the value of a (9, a) 0 1 x (ii) Sketch the graph of f(x + 2) + 1

24 Transformation of log graphs 2. 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. Daily Practice

25 Today we will be working out questions on log graphs ad mixed log questions. Homework Due Tuesday Specimen paper

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27 2012