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Lesson 5-5 Logarithms

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Logarithmic functions

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The inverse of the exponential function.

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Logarithmic functions The inverse of the exponential function. Basic exponential function: f(x) = b x

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Logarithmic functions The inverse of the exponential function. Basic exponential function: f(x) = b x

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Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x

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Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x

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Logarithmic functions The inverse of the exponential function. Basic logarithmic function: f -1 (x) = log b x Every (x,y) (y,x)

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Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa):

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Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a b a = x

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Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a b a = x The base of the logarithmic form becomes the base of the exponential form.

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Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a b a = x The answer to the log statement becomes the power in the exponential form.

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Logarithmic functions Basic rule for changing exponential equations to logarithmic equations (or vice-versa): log b x = a b a = x The number you are to take the log of in the log form, becomes the answer in the exponential form.

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Examples:

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log 5 25 = 2 because 5 2 = 25

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Examples: log 5 25 = 2 because 5 2 = 25 log 5 125 = 3 because 5 3 = 125

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Examples: log 5 25 = 2 because 5 2 = 25 log 5 125 = 3 because 5 3 = 125 log 2 (1/8) = - 3 because 2 -3 = 1/8

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base b exponential function f(x) = b x

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base b exponential function f(x) = b x Domain: All reals Range: All positive reals

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base b logarithmic function f -1 (x) = log b (x)

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base b logarithmic function f -1 (x) = log b (x) Domain: All positive reals Range: All reals

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Types of Logarithms

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There are two special logarithms that your calculator is programmed for:

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Types of Logarithms There are two special logarithms that your calculator is programmed for: log 10 (x) called the common logarithm

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Types of Logarithms There are two special logarithms that your calculator is programmed for: log 10 (x) called the common logarithm For the common logarithm we do not include the subscript 10, so all you will see is: log (x)

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Types of Logarithms There are two special logarithms that your calculator is programmed for: So, log 10 (x) log (x) = k if 10 k = x

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Types of Logarithms There are two special logarithms that your calculator is programmed for: log e (x) called the natural logarithm

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Types of Logarithms There are two special logarithms that your calculator is programmed for: log e (x) called the natural logarithm For the natural logarithm, we do not include the subscript e, so all you will see is: ln (x)

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Types of Logarithms There are two special logarithms that your calculator is programmed for: So, log e (x) ln (x) = k if e k = x

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Examples:

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log 6.3 = 0.8 because 10 0.8 = 6.3

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Examples: log 6.3 = 0.8 because 10 0.8 = 6.3 ln 5 = 1.6 because e 1.6 = 5

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Example:

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Find the value of x to the nearest hundredth.

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Example: Find the value of x to the nearest hundredth.

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Example: Find the value of x to the nearest hundredth. 10 x = 75

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Example: Find the value of x to the nearest hundredth. 10 x = 75 This transfers to the log statement log 10 75 = x and the calculator will tell you x = 1.88

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Example: Find the value of x to the nearest hundredth. e x = 75

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Example: Find the value of x to the nearest hundredth. e x = 75 This transfers to the log statement ln 75 = x and the calculator will tell you x = 4.32

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Evaluate:

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Solve:

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Assignment: Pg. 194 C.E. #1 – 9 all W.E. #2 – 14 evens

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