Presentation on theme: "5.4 Common and Natural Logarithmic Functions"— Presentation transcript:
15.4 Common and Natural Logarithmic Functions Do NowSolve for x.1. 5x= x=23. 3x= x=130
25.4 Common and Natural Logarithmic Functions Do NowSolve for x.1. 5x=25 x= x= x= ½3. 3x=27 x= x=130 x≈2.11
3Common LogarithmsThe inverse function of the exponential function f(x)=10x is called the common logarithmic function.Notice that the base is 10 – this is specific to the “common” logThe value of the logarithmic function at the number x is denoted as f(x)=log x.The functions f(x)=10x and g(x)=log x are inverse functions.log v = u if and only if 10u = vNotice that the base is “understood “to be 10.Because exponentials and logarithms are inverses of one another, what do we know about their graphs?
4Common LogarithmsSince logs are a special kind of exponent, each logarithmic statement can be expressed as an exponential.LogarithmicExponentiallog 29 == 29log 378 == 378
5Example 1: Evaluating Common Logs Without using a calculator, find each value.log 1000log 1log 10log (-3)
6Example 1: Solutions Without using a calculator, find each value log 1000 10x = 1000 log 1000 = 3log 1 10x = 1 log 1 = 0log 10 10x = 10 log 10 = 1/2log (-3) 10x = -3 undefined
7Evaluating Logarithms A calculator is necessary to evaluate most logs, but you can get a rough estimate mentally.For example, because log 795 is greater than log 100 = 2 and less than log 1000 = 3, you can estimate that log 795 is between 2 and 3, and closer to 3.
8Using Equivalent Statements A method for solving logarithmic or exponential equations is to use equivalent exponential or logarithmic statements.For example:To solve for x in log x = 2, we can use 102 = x and see that x = 100To solve for x in 10x = 29, we can use log 29 = x, and using a calculator to evaluate shows that x =
9Example 2: Using Equivalent Statements Solve each equation by using an equivalent statement.log x = 510x = 52
10Example 2: SolutionSolve each equation by using an equivalent statement.log x = = x x = 100,00010x = 52 log 52 = x x ≈
11Natural LogarithmsThe exponential function f(x)=ex is useful in science and engineering. Consequently, another type of logarithm exists, where the base is e instead of 10.The inverse function of the exponential function f(x)=ex is called the natural logarithmic function.The value of this function at the number x is denoted as f(x)=ln x and is called the natural logarithm.
12Natural LogarithmsThe functions f(x)=ex and g(x)=ln x are inverse functions.ln v = u if and only if eu = vNotice that the base is “understood” to be e.Again, as with common logs, every natural logarithmic statement is equivalent to an exponential statement.LogarithmicExponentialln 14 =e = 14ln 0.2 =e = 0.2
13Example 3: Evaluating Natural Logs Use a calculator to find each valueln 1.3ln 203ln (-12)
14Example 3: Solutions Use a calculator to find each value ln 1.3 .2624 ln (-12) undefinedWhy is this undefined??
15Example 4: Solving by Using and Equivalent Statement Solve each equation by using an equivalent statement.ln x = 2ex = 8
16Example 4: SolutionsSolve each equation by using an equivalent statement.ln x = 2 e2 = x x =ex = 8 ln8 = x x =
17Graphs of Logarithmic Functions The following table compares graphs of exponential and logarithmic functions (page 359 in your text):Exponential FunctionsLogarithmic FunctionsExamplesf(x) = 10x; f(x) = exg(x) = log x; g(x) = ln xDomainAll real numbersAll positive real numbersRangef(x) increases as x increasesg(x) increases as x increasesf(x) approaches the x-axis as x-decreasesg(x) approaches the y-axis as x approaches 0Reference Point(0, 1)(1, 0)
18Example 5: Transforming Logarithmic Functions Describe the transformation of the graph for each logarithmic function. Identify the domain and range.3log(x+4)ln(2-x)-3
19Example 5: Transforming Logarithmic Functions Describe the transformation of the graph for each logarithmic function. Identify the domain and range.3log(x+4)Shifted to the left 4 units; vertically stretched by 3Domain: x > -4 Range: All real numbersln(2-x)-3 = ln(-(x-2))-3Horizontal reflection across y-axis; 2 units to theright; 3 units downDomain: x > 2 Range: All real numbers