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Logarithms and Their Properties

Logarithms and Their Properties. Lesson 4.1. Recall the Exponential Function. General form Given the exponent what is the resulting y-value? Now we look at the inverse of this function Now we will ask, given the result, what exponent is needed to achieve it?. A New Function.

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Logarithms and Their Properties

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  1. Logarithms and Their Properties Lesson 4.1

  2. Recall the Exponential Function • General form • Given the exponentwhat is the resulting y-value? • Now we look at the inverse of this function • Now we will ask, given the result, what exponent is needed to achieve it?

  3. A New Function • Consider the exponential function y = 10x • Based on that function, declare a new function x = log10y • You should be able to see that these are inverse functions • In general • The log of a numberis an exponent

  4. Note: if no base specified, default is base of 10 The Log Function • Try Theselog39 = ? log232 = ? log 0.01 = ?

  5. Properties of Logarithms • Note box on page 154 of text • Most used properties

  6. Natural Logarithms • We have used base of 10 for logs • Another commonly used base for logs is e • e is an irrational number (as is ) • e has other interesting properties • Later to be discovered in calculus • Use ln button on your calculator

  7. Properties of the Natural Logarithm • Recall that y = ln x  x = ey • Note that • ln 1 = 0 and ln e = 1 • ln (ex) = x (for all x) • e ln x = x (for x > 0) • As with other based logarithms

  8. Note this is not the same aslog 1.04 – log 3 Use Properties for Solving Exponential Equations • Given • Take log ofboth sides • Use exponent property • Solve for whatwas the exponent

  9. Misconceptions • log (a+b) NOT the same as log a + log b • log (a-b) NOT the same as log a – log b • log (a * b) NOT same as (log a)(log b) • log (a/b) NOT same as (log a)/(log b) • log (1/a) NOT same as 1/(log a)

  10. Assignment • Lesson 4.1 • Page 157 • Exercises 1 – 51 odd

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