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9.1, 9.3 Exponents and Logarithms

9.1, 9.3 Exponents and Logarithms. 24. Graphs of Exponential Functions. The graph of f(x) = b x has a characteristic shape. If b > 1, the graph goes uphill If 0 < b < 1, the graph goes downhill Domain is (– ∞, ∞). Range is (0, ∞)

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9.1, 9.3 Exponents and Logarithms

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  1. 9.1, 9.3 Exponents and Logarithms

  2. 24 Graphs of Exponential Functions • The graph of f(x) = bx has a characteristic shape. • If b> 1, the graph goes uphill • If 0 < b< 1, the graph goes downhill • Domain is (–∞, ∞). Range is (0, ∞) • Unless translated the graph has a y-intercept of (0,1)

  3. Definition of a Logarithm • A logarithm, or log, is defined in terms of an exponent: • If 52=25 then log525=2 • You can say that the log is the exponent we put on 5 to get 25 • If bx=a, then logba=x

  4. Logarithmic Functions Logarithmic Equation y = log2x x = 2yis anexponentialequation. If we solved for “y” we would get alogarithmic equation. Here are the parts of each type of equation: Exponential Equationx = 2y exponent /logarithm base number

  5. Rewrite in exponential form! loga64 = 2 base number exponent Example: Solve loga64 = 2 a2 = 64 a = + 8 → a = 8 Example : Solve log5x = 3 Rewrite in exponential form: 53 = x x = 125

  6. Re-write it as an exponential function and make a T-chart: Rewrite as:x = 3y How do you graph a logarithmic function? y = 3x xy 1/9 1/3 1 3 9 -2 -1 0 1 2 Example: Graph y = log3x y = log3 x

  7. -1 1 2 3 4 5 6 Graphs of Logarithmic Functions • The graph of f(x)=logbx has a characteristic shape. • The domain of the function is {x | x >0} • Unless translated, the graph has an x-intercept of 1. • Note the domain and range!

  8. The logarithm with base 10 is called the common logarithm (this is the one your calculator evaluates with the LOG button) The logarithm with base e is called the natural logarithm (this is the one your calculator evaluates with the LN button)

  9. Examples. Evaluate each:a. log8 84b. 6[log6 (3y – 1)] logb bx = x log8 84 = 4 blogb x = x 6[log6 (3y – 1)]= 3y – 1 Here are some IMPORTANT logarithm properties: 1. loga 1 = 0 because a0 = 1 2. loga a = 1 because a1 = a 3. loga ax = x because ax = ax

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