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5.4 Common and Natural Logarithmic Functions Do Now Solve for x x = x = x = x =130

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5.4 Common and Natural Logarithmic Functions Do Now Solve for x x =25x=22. 4 x =2 x= ½ 3. 3 x =27x= x =130 x2.11

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Common Logarithms The inverse function of the exponential function f(x)=10 x 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)=10 x and g(x)=log x are inverse functions. log v = u if and only if 10 u = 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?

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Since logs are a special kind of exponent, each logarithmic statement can be expressed as an exponential. Common Logarithms LogarithmicExponential log 29 = = 29 log 378 = = 378

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Example 1: Evaluating Common Logs Without using a calculator, find each value. 1.log log 1 3.log 10 4.log (-3)

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Example 1: Solutions Without using a calculator, find each value 1.log x = 1000 log 1000 = 3 2.log 1 10 x = 1 log 1 = 0 3.log x = 10 log 10 = 1/2 4.log (-3) 10 x = -3 undefined

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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.

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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 10 2 = x and see that x = 100 – To solve for x in 10 x = 29, we can use log 29 = x, and using a calculator to evaluate shows that x =

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Example 2: Using Equivalent Statements Solve each equation by using an equivalent statement. 1.log x = x = 52

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Example 2: Solution Solve each equation by using an equivalent statement. 1.log x = = x x = 100, x = 52 log 52 = x x

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Natural Logarithms The exponential function f(x)=e x 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)=e x 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.

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Natural Logarithms The functions f(x)=e x and g(x)=ln x are inverse functions. ln v = u if and only if e u = 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. LogarithmicExponential ln 14 = e = 14 ln 0.2 = e = 0.2

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Example 3: Evaluating Natural Logs Use a calculator to find each value 1.ln ln ln (-12)

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Example 3: Solutions Use a calculator to find each value 1.ln ln ln (-12)undefined Why is this undefined??

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Example 4: Solving by Using and Equivalent Statement Solve each equation by using an equivalent statement. 1.ln x = 2 2.e x = 8

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Example 4: Solutions Solve each equation by using an equivalent statement. 1.ln x = 2e 2 = x x = e x = 8ln8 = x x =

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Graphs of Logarithmic Functions The following table compares graphs of exponential and logarithmic functions (page 359 in your text): 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 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)

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Example 5: Transforming Logarithmic Functions Describe the transformation of the graph for each logarithmic function. Identify the domain and range. 1.3log(x+4) 2.ln(2-x)-3

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Example 5: Transforming Logarithmic Functions Describe the transformation of the graph for each logarithmic function. Identify the domain and range. 1.3log(x+4) Shifted to the left 4 units; vertically stretched by 3 Domain: x > -4Range: All real numbers 2.ln(2-x)-3 = ln(-(x-2))-3 Horizontal reflection across y-axis; 2 units to the right; 3 units down Domain: x > 2Range: All real numbers

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