5.4 Common and Natural Logarithmic Functions Do Now Solve for x. 1. 5x=25 2. 4x=2 3. 3x=27 4. 10x=130

5.4 Common and Natural Logarithmic Functions Do Now Solve for x. 1. 5x=25 x=2 2. 4x=2 x= ½ 3. 3x=27 x=3 4. 10x=130 x≈2.11

Common Logarithms The 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” log The 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 = v Notice that the base is “understood “to be 10. Because exponentials and logarithms are inverses of one another, what do we know about their graphs?

Common Logarithms Since logs are a special kind of exponent, each logarithmic statement can be expressed as an exponential. Logarithmic Exponential log 29 = 1.4624 101.4624 = 29 log 378 = 2.5775 102.5775 = 378

Example 1: Evaluating Common Logs Without using a calculator, find each value. log 1000 log 1 log 10 log (-3)

Example 1: Solutions Without using a calculator, find each value log 1000  10x = 1000  log 1000 = 3 log 1  10x = 1  log 1 = 0 log 10  10x = 10  log 10 = 1/2 log (-3)  10x = -3  undefined

Evaluating 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.

Using 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 = 100 To solve for x in 10x = 29, we can use log 29 = x, and using a calculator to evaluate shows that x = 1.4624

Example 2: Using Equivalent Statements Solve each equation by using an equivalent statement. log x = 5 10x = 52

Example 2: Solution Solve each equation by using an equivalent statement. log x = 5 105 = x  x = 100,000 10x = 52 log 52 = x  x ≈ 1.7160

Natural Logarithms The 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.

Natural Logarithms The functions f(x)=ex and g(x)=ln x are inverse functions. ln v = u if and only if eu = v Notice that the base is “understood” to be e. Again, as with common logs, every natural logarithmic statement is equivalent to an exponential statement. Logarithmic Exponential ln 14 = 2.6391 e2.6391 = 14 ln 0.2 = -1.6094 e-1.6094 = 0.2

Example 3: Evaluating Natural Logs Use a calculator to find each value ln 1.3 ln 203 ln (-12)

Example 3: Solutions Use a calculator to find each value ln 1.3 .2624 ln (-12) undefined Why is this undefined??

Example 4: Solving by Using and Equivalent Statement Solve each equation by using an equivalent statement. ln x = 2 ex = 8

Example 4: Solutions Solve each equation by using an equivalent statement. ln x = 2 e2 = x  x = 7.3891 ex = 8 ln8 = x  x = 2.0794

Graphs of Logarithmic Functions The following table compares graphs of exponential and logarithmic functions (page 359 in your text): Exponential Functions Logarithmic Functions Examples f(x) = 10x; f(x) = ex g(x) = log x; g(x) = ln x Domain All real numbers All positive real numbers Range f(x) increases as x increases g(x) increases as x increases f(x) approaches the x-axis as x-decreases g(x) approaches the y-axis as x approaches 0 Reference Point (0, 1) (1, 0)

Example 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

Example 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 3 Domain: x > -4 Range: All real numbers ln(2-x)-3 = ln(-(x-2))-3 Horizontal reflection across y-axis; 2 units to the right; 3 units down Domain: x > 2 Range: All real numbers