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Chapter 8:Exponential and Logarithmic Functions

Chapter 8:Exponential and Logarithmic Functions. Presentation By: Jasmit , Samir , and Raj. Getting Started. Skills you need for this chapter: Knowledge of the 3 exponent laws (Ex. ( x a ) ( x b ) = x a+b Know how to graph exponential functions and their properties.

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Chapter 8:Exponential and Logarithmic Functions

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  1. Chapter 8:Exponential and Logarithmic Functions Presentation By: Jasmit, Samir, and Raj

  2. Getting Started Skills you need for this chapter: • Knowledge of the 3 exponent laws (Ex. (xa) (xb) = xa+b • Know how to graph exponential functions and their properties. • How to find the inverse of any function. • Use exponential functions to model word problems.

  3. 8.1 Exploring the Logarithmic Function • It is the inverse of the exponential function y=ax. It can be defined by x=ay or y=logax • Log means “the exponent that must be applied to base a to get the value of x” Properties: • Since the x-axis is the HA for the exponential function, the y-axis becomes the VA for the logarithmic function • The x-intercept of the logarithmic function is the y-intercept of the exponential function, therefore it is 1. • The Domain of y=logax is {x E R | x> 0} and the Range is {y E R} NOTE: they are the opposite of the Domain and Range of y=ax.

  4. 8.1 continued Note: The base “a” in y=logax determines the shape of the function. If a < 1, the function will always increasing. If 0 > a > 1, the function will always be decreasing. Note : if a > 0, and x is a negative value, it is impossible to evaluate. For example log3(-9)=?  3x=-9 Impossible a<1 VS. 0>a>1

  5. 8.2 Transformations of Logarithmic Functions All transformations that can be applied to this graph are defined by the following equation: f(x)=alog10 [k(x-d)]+c a: vertical stretches and reflections on the x-axis. k: 1/k gives the vertical stretch, and any reflections on the y-axis. d gives the horizontal shift and c gives the vertical shift. The VA is always at “d” [for example y=log (x-2), VA at x=2] if the function is to the left of the VA, the domain is x< d; Domain of a function on the right of the VA is x>d.

  6. Let’s practice  1. Write the inverse of y=4x in exponential for AND logarithmic form. 2. f(x)=log10x is stretched vertically by 3, reflected in the x-axis, horizontally stretched by 2, horizontally shifted 5 right, and vertically translated 2 units up. Determine a) the new equation, and b) the new domain and range. 2 BRAVE volunteers please?

  7. 8.3 Solving Exponential Equations Ex. 1 3x = 9 3x=32 x=2 Ex. 2 3x=9 xlog3=log9 x=log9/log3 x=2 3 ways to evaluate an exponential equation: • express both sides with a common base, then make both exponents equal one another and solve.   • graph both sides of the equal sign, then determining the point of intersection. (recommended only if you have a graphing calculator) • rewrite in log form then simplify.

  8. Do Not Forget! Base Conversion Formula: logab=log10b log10a NOTE:USEFUL WHEN USING YOUR CALCULATOR TO SOLVE 1. There are no logarithms of negative numbers, as negative numbers cannot be expressed as powers of positive bases. • Ex: log-9=x not possible 2. log x is always a logarithm with a base of 10 3. Properties: • loga1 = 0 • logaax = x • aloga X = x

  9. 8.4 Laws of Logarithms Example: log 5 + log 6- log 3 =log(5)(6)/(3) =log10 =1 The laws of logarithms are the same as those of exponents because logarithms are actually exponents. NOTE: these laws can only be used if the base is the same! Product Law: log x+ log y= logxy Quotient Law: log x- log y= log (x/y) Power Law: log xy= y log x

  10. Some more practice… 1. a) log381=? b) log2/327/8=? 2. Evaluate. a)log1133 – log113 b) log4 3√16

  11. 8.5 Solving Exponentials Ex: 3X-3=32 x-3=2 x=5 If x=y, then log x= log y Ex: 3x=9 Log3x=log9 Xlog3=log9 X=log9/log3 x=2 If two exponential expressions with the same base are equal, then there exponents are equal. If two expressions are equal, then taking of the log of both sides still keeps them equal.

  12. 8.5 continued 2x=10 xlog2=log10 x=log10/log2 x=3.32 3 New Equations: Compound Interest A=AO(1+i)n Half-Life A=Ao(1/2) t/h Double Life A=AO(2)t/h When Evaluating Algebraically Keep in Mind: Take the logarithm of both sides, then isolate for the unknown value. Sometimes, you can set both sides of the equal sign to the same base. The exponents will now be equal, allowing you to solve easily.

  13. 8.6 Solving Logarithmic Equations Ex. Logx27=3/2 X3/2=27 X=272/3 X=9 Log b(x)=y where b>0, b cant be 1 And x>0  Simplest way to solve: express the logarithm in exponential form, and solve the equation.  When solving logarithms remember to use the 3 laws to help simplify (the product, quotient, and power laws) *NOTE: remember that the base and what’s inside the logarithm (called an argument) must both be positive!

  14. 8.7 Solving Problems with Exp and Log functions R=log (I/Io) pH=-log [H+] • Logarithmic scales are used for a variety of measurements in the real world: 1. Richter scale: used to measure the intensity of earthquakes 2. pH Scale: used to measure acidity 3. Sound intensity: L=10log(I/Io) where Io is 10-12

  15. We Saved the Best for Last VERSUS The Audi The Aston Martin Jasmit, Raj, and Samir need a faster car so they can get to school as soon as possible to help students with Chapter 8. Jasmit insists on the Aston Martin db9 ($80 000) but Samir and Raj say the Audi r8 ($70 000). They have $50 000 saved in their expert group savings account and they ask Mrs. Rishad to help them decide. If the account pays 10%/a compounded annualy for 4.5 years years, which car should they buy?

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