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Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions Essential Question: What is the relationship between a logarithm.

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Presentation on theme: "Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions Essential Question: What is the relationship between a logarithm."— Presentation transcript:

1 Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions Essential Question: What is the relationship between a logarithm and an exponent?

2 5.4: Common and Natural Logarithmic Functions You’ve ran across a multitude of inverses in mathematics so far... ◦ Additive Inverses: 3 & -3 ◦ Multiplicative Inverses: 2 & ½ ◦ Inverse of powers: x 4 & or x ¼ ◦ But what do you do when the exponent is unknown? For example, how would you solve 3 x = 28, other than guess & check? ◦ Welcome to logs…

3 5.4: Common and Natural Logarithmic Functions Logs ◦ There are three types of commonly used logs  Common logarithms (base 10)  Natural logarithms (base e)  Binary logarithms (base 2) ◦ We’re only going to concentrate on the first two types of logarithms, the 3 rd is used primarily in computer science. ◦ Want to take a guess as to why I used the words “base” above?

4 5.4: Common and Natural Logarithmic Functions Common logarithms ◦ The functions f(x) = 10 x and g(x) = log x are inverse functions  log v = u if and only if 10 u = v ◦ All logs can be thought of as a way to solve for an exponent  Log base answer = exponent ◦ log 10 2 x = x 2 x =

5 5.4: Common and Natural Logarithmic Functions Common logarithms ◦ Scientific/graphing calculators have the logarithmic tables built in, on our TI-86s, the “log” button is below the graph key. ◦ To find the log of 29, simply type “log 29”, and you will be returned the answer  That means, = 29 ◦ Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

6 5.4: Common and Natural Logarithmic Functions Evaluating Common Logarithms ◦ Without using a calculator, find the following ◦ log 1000  ◦ log 1  ◦ log  ◦ log (-3)  If log 1000 = x, then 10 x = Because 10 3 = 1000, log 1000 = 3 If log 1 = x, then 10 x = 1 Because 10 0 = 1, log 1 = 0 If log (-3) = x, then 10 x = (-3) Because there is no real number exponent of 10 to get -3 (or any negative number, for that matter), log(-3) is undefined

7 5.4: Common and Natural Logarithmic Functions Using Equivalent Statements (log) ◦ Solve each by using equivalent statements (and calculator, if necessary) ◦ log x = 2  ◦ 10 x = 29  ◦ Remember  Log base answer = exponent log x = 2 → 10 2 = x → 100 = x 10 x = 29 → log 29 = x → = x

8 5.4: Common and Natural Logarithmic Functions Natural logarithms ◦ (or Captain’s Log, star date …)  Common logarithms are used when the base is 10.  Another regular base is used with exponents, that being the irrational constant e. ◦ For natural logarithms, we use “ln” instead of “log”. The ln key is located beneath the log key on your calculator.

9 5.4: Common and Natural Logarithmic Functions Evaluating Natural Logarithms ◦ Use a calculator to find each value.  ln 0.15  ln 0.15 = , which means e = 0.15  ln 186  ln 186 = , which means e = 186  ln (-5)  Undefined, as it’s not possible for a positive number (e) to somehow yield a negative number.

10 5.4: Common and Natural Logarithmic Functions Using Equivalent Statements (ln) ◦ Solve each by using equivalent statements (and calculator, if necessary) ◦ ln x = 4  ◦ e x = 5  ◦ Remember  Log base answer = exponent ln x = 4 → e 4 = x → = x e x = 5 → ln 5 = x → = x

11 5.4: Common and Natural Logarithmic Functions Assignment ◦ Page 361, 2 – 36 (even problems)  Even problems are done exactly like the odd problems, which are in the back of the book)

12 Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions Day 2 Essential Question: What is the relationship between a logarithm and an exponent?

13 5.4: Common and Natural Logarithmic Functions Graphs of Logarithmic Functions Exponential functionsLogarithmic functions Examplesf(x) = 10 x ; f(x) = e x g(x) = log x; g(x) = ln x DomainAll real numbersAll positive real numbers RangeAll positive real numbersAll real numbers Otherf(x) increases as x increasesg(x) increases as x increases f(x) approaches the x-axis as x decreases g(x) approaches the y-axis as x approaches 0

14 5.4: Common and Natural Logarithmic Functions Transforming Logarithmic Functions ◦ Same as before…  Changes next to the x affect the graph horizontally and opposite as would be expected  Changes away from the x affect the graph vertically and as expected  Example  Describe the transformation from the graph of g(x) = log x to the graph of h(x) = 2 log (x – 3). Give the domain and range.  Vertical stretch by a factor of 2  Horizontal shift to the right 3 units  Domain: The domain of a log function is all positive real numbers (x > 0). Shifting three units right means the new domain is x > 3.  Range: The range of a log function is all real numbers. That doesn’t change by transforming the graph.

15 5.4: Common and Natural Logarithmic Functions Transforming Logarithmic Functions ◦ Example #2  Describe the transformation from the graph of g(x) = ln x to the graph of h(x) = ln (2 – x) - 3. Give the domain and range.  x is supposed to come first, so h(x) should be rewritten as h(x) = ln [-(x – 2)] - 3  Horizontal reflection  Horizontal shift to the right 2 units  Vertical shift down 3 units  Domain: The domain of a log function is all positive real numbers (x > 0). The horizontal reflection flips the sign, and shifting two units right means the new domain is x < 2.  Range: The range of a log function is all real numbers. That doesn’t change by transforming the graph.

16 5.4: Common and Natural Logarithmic Functions Assignment ◦ Page 361, 37 – 48 (all problems)  Problems 37 – 40 only ask to find the domain, but you may need to figure out the translation first.  Even problems are done exactly like the odd problems, which are in the back of the book)


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