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LOGS EQUAL THE

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The inverse of an exponential function is a logarithmic function. Logarithmic Function x = log a y read: “x equals log base a of y”

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y = b x x = log b y These two equations are equivalent We can convert exponential equations to logarithmic equations and vice versa, using this:

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Convert to exponential form 1) 2) 3)

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Convert to logarithmic form 4) 5) 6)

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Now that we can convert between the two forms we can simplify logarithmic expressions. Without a Calculator!

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Simplify 1) log 2 32 2) log 3 27 3) log 4 2 4) log 3 1 2 ? = 32 3 ? = 27 4 ? = 2 3 ? = 1 ? = 5 ? = 3 ? = 0.5 ? = 0 “What is the exponent of that gives you 32?” “ What is the exponent of 3 that gives you 27? ”

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Evaluate

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We can also use these two forms to help us solve for an inverse. The steps for finding an inverse are the same as before. Easy as 1, 2, 3… 1-Rewrite 2-Switch x and y 3-Solve for y

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Example: Find the inverse

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Now you try……..

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An inverse you just have to know Ln and are inverses They undo each other 1. 2.

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Example: Find the inverse

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Now you try….. Find the inverse:

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Essential Question: How do I graph & solve exponential and logarithmic functions? Daily Question: How do you expand and condense logs?

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Change-of-Base Formula Let a, b, and x be positive real numbers such that a 1 and b 1.

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Ex. 1 a) Evaluate using the change-of-base formula. Round to four decimal places.

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Ex. 2 You can do the same problem using natural logarithms. a) Evaluate using the natural logarithms. Round to four decimal places.

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Properties of Logarithms Product Property Quotient Property Power Property

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Ex. 3 log 10 5x 3 y log 10 5+ log 10 y+ log 10 x 3 log 10 5+ log 10 y+ 3 log 10 x Expand.

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Ex. 4 Expand.

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Now you Try….. A.A. B.C. Expand each log 3 2x 6 y

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Ex. 5 Condense. a) b)

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Now you Try…..Condense each

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Ex. 6 Use and to evaluate the logarithm.

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Ex. 7 Use the properties of logarithms to verify that -ln ½ = ln 2 -ln ½ = -ln (2 -1 ) = -(-1) ln (2) = ln (2) =ln 2

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Homework: Page 147 # 1 – 23 odd Page 157 # 1 – 25 odd

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