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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 and an exponent?

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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…

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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?

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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 =

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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 1.4624. That means, 10 1.4624 = 29 ◦ Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

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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 = 1000. 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

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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 → 1.4624 = x

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5.4: Common and Natural Logarithmic Functions Natural logarithms ◦ (or Captain’s Log, star date 2.71828182846…) 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.

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5.4: Common and Natural Logarithmic Functions Evaluating Natural Logarithms ◦ Use a calculator to find each value. ln 0.15 ln 0.15 = -1.8971, which means e -1.8971 = 0.15 ln 186 ln 186 = 5.2257, which means e 5.2257 = 186 ln (-5) Undefined, as it’s not possible for a positive number (e) to somehow yield a negative number.

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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 → 54.5982 = x e x = 5 → ln 5 = x → 1.6094 = x

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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)

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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?

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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

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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.

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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.

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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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