Exponential/ Logarithmic

Exponential/ Logarithmic

Exponential/ Logarithmic Exponential Functions f(x) = ax Domain (-, ) Three types:

1) if 0 < a < 1 2) if a = 1 3) if a > 1 Range (0, )

Laws of Exponents a x + y = ax ay ax/ ay = a x y (ax)y = axy (ab)x = axbx

Sketching Example Sketch the function y = 3 2x 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 Example Find: log10(0.001)

log216 Log Graph Reflection of exponential function about the line y = x Domain (0, )

Range (-,) Laws of Logarithms loga(xy) = logax + logay loga(x/y) = logax logay

logaxr = rlogax Example Evaluate log280 log25 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 = 2.71828 Example Graph y = e-x 1 and find the domain and range

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

Properties of Natural Logs ln(ex) = x elnx = x ln e = 1

Example Find x if lnx = 5 Example

Solve e5 3x = 10 Example Express ln a + ln b as a single

logarithm 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 Example

Evaluate log85 Example

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

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