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**Exponential/ Logarithmic**

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**Exponential Functions**

f(x) = ax Domain (-∞, ∞) Range (0, ∞) Three types: 1) if 0 < a < 1 2) if a = 1 3) if a > 1

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**Laws of Exponents a x + y = ax ay ax/ ay = a x –y (ax)y = axy**

(ab)x = axbx

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Sketching Example Sketch the function y = 3 – 2x

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**Exponential Functions are One to One**

Has an inverse f-1 which is called the logarithmic function (loga) f-1(x) = y f(y) = x ay = x logax = y

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Example Find: log10(0.001) log216

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**Log Graph Reflection of exponential function about the line y = x**

Domain (0, ∞) Range (-∞,∞)

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**Laws of Logarithms loga(xy) = logax + logay loga(x/y) = logax – logay**

logaxr = rlogax

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Example Evaluate log280 – log25

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e y = ax Many formulas in calculus are greatly simplified if we use a base a such that the slope of the tangent line at y = 1 is exactly 1 For y = 2x, slope at y = 1 is .7 For y = 3x, slope at y = 1 is 1.1 Value of a lies between 2 and 3 and is denoted by the letter e e =

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Example Graph y = ½ e-x – 1 and find the domain and range

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Natural log (ln) Log with a base of e logex = lnx lnx = y ey = x

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**Properties of Natural Logs**

ln(ex) = x elnx = x ln e = 1

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Example Find x if lnx = 5

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Example Solve e5 – 3x = 10

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Example Express ln a + ½ ln b as a single logarithm

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**Expression y = logax ay = x ln ay = ln x y ln a = ln x y = ln x/ ln a**

logax = ln x/ ln a if a ≠ 0

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Example Evaluate log85

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Example The half-life of a radioactive substance given by f(t) = 24 ∙ 2-t/25 Find the inverse

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