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Logarithmic Functions

Logarithmic Functions. Argument. Exponent. Base. y = log a x , is read “the logarithm, base a , of x ,” or “log, base a , of x ,” means “the exponent to which we raise a to get x .”. Is equivalent to. Example. Simplify:. a) log 3 81 b) log 3 1 c) log 3 (1/9).

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Logarithmic Functions

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  1. Logarithmic Functions

  2. Argument Exponent Base y = logax, is read “the logarithm, base a, of x,” or “log, base a, of x,” means “the exponent to which we raise a to get x.” Is equivalent to

  3. Example Simplify: a) log381 b) log31 c) log3(1/9) Solution a) Think of log381 as the exponent to which we raise 3 to get 81. That exponent is 4. Therefore, log381 = 4. b) We ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. Thus, log31 = 0. c) To what exponent do we raise 3 in order to get 1/9? Since 3-2 = 1/9, we have log3(1/9) = –2.

  4. Simplify: Example Solution Remember that log523 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, we have It is important to remember that a logarithm is an exponent.

  5. Example Graph y = f (x) = log3x. Solution

  6. Common Logarithms Base-10 logarithms, called common logarithms, are useful because they have the same base as our “commonly” used decimal system, and it is one of two logarithms on our calculator. We’ll discuss this later. Example

  7. Example Graph: y = log (x/4) – 2 in the window [-2, 8] X [-5,5]. Solution

  8. Equivalent Equations We use the definition of logarithm to rewrite a logarithmic equation as an equivalent exponential equation or the other way around: m = logaxis equivalent to am = x.

  9. Example Rewrite each as an equivalent exponential equation: a) –m = log3x b) 6 = logaz The logarithm is the exponent. The base remains the base. Solution a) –m = log3x is equivalent to 3-m= x b) 6 = logaz is equivalent to a6 = z.

  10. Example Rewrite each as an equivalent logarithmic equation: a) 49 = 7x b) x -2 = 9 The exponent is thelogarithm. The base remains the base. Solution a) 49 = 7x is equivalent to x = log749 b) x -2 = 9 is equivalent to –2= logx9.

  11. Solving Certain Logarithmic Equations Logarithmic equations are often solved by rewriting them as equivalent exponential equations.

  12. Example Solve: a) log3x = –3; b) logx4 = 2. Solution a) log3x = –3 x = 3–3 = 1/27 The solution is 1/27. The check is left to the student. b) logx4 = 2 4 = x2 x = 2 or x = –2 Because all logarithmic bases must be positive, –2 cannot be a solution. The solution is 2.

  13. Example Solve: a) log636 = x; b) log91 = t. Solution a) log636 = x b) log91 = t 6x = 36 9t = 1 6x = 62 9t = 90 x = 2 t = 0

  14. = 0 = 1

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