Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions

Chapter 5: Exponential and Logarithmic Functions5.4: Common and Natural Logarithmic Functions Essential Question: What is the relationship between a logarithm and an exponent?

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: x4 & or x¼ • But what do you do when the exponent is unknown? For example, how would you solve 3x = 28, other than guess & check? • Welcome to logs…

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 3rd is used primarily in computer science. • Want to take a guess as to why I used the words “base” above?

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

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, 101.4624 = 29 • Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

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 10x = 1000. Because 103 = 1000, log 1000 = 3 If log 1 = x, then 10x = 1 Because 100 = 1, log 1 = 0 If log (-3) = x, then 10x = (-3) Because there is no real number exponent of 10 to get -3 (or any negative number, for that matter), log(-3) is undefined

5.4: Common and Natural Logarithmic Functions • Using Equivalent Statements (log) • Solve each by using equivalent statements (and calculator, if necessary) • log x = 2 • 10x = 29 • Remember • Log base answer = exponent log x = 2 → 102 = x → 100 = x 10x = 29 → log 29 = x → 1.4624 = x

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.

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 e5.2257 = 186 • ln (-5) • Undefined, as it’s not possible for a positive number (e) to somehow yield a negative number.

5.4: Common and Natural Logarithmic Functions • Using Equivalent Statements (ln) • Solve each by using equivalent statements (and calculator, if necessary) • ln x = 4 • ex = 5 • Remember • Log base answer = exponent ln x = 4 → e4 = x → 54.5982 = x ex = 5 → ln 5 = x → 1.6094 = x

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)

Chapter 5: Exponential and Logarithmic Functions5.4: Common and Natural Logarithmic FunctionsDay 2 Essential Question: What is the relationship between a logarithm and an exponent?

5.4: Common and Natural Logarithmic Functions • Graphs of Logarithmic Functions

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.

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.

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)

Chapter 5: Exponential and Logarithmic Functions 5.4: Common and Natural Logarithmic Functions