## Presentation on theme: "Logarithmic Functions y = log a x, is read “the logarithm, base a, of x,” or “log, base a, of x,” means “the exponent to lớn which we raise a to get x.”"— Presentation transcript:

Bạn đang xem: Logarithmic functions 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

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2 Logarithmic Functions

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

5 Example Solution a) log 3 81 b) log 3 1 c) log 3 (1/9) a) Think of log 3 81 as the exponent khổng lồ which we raise 3 khổng lồ get 81. That exponent is 4. Therefore, log 3 81 = 4. Simplify: b) We ask: “To what exponent bởi vì we raise 3 in order to lớn get 1?” That exponent is 0. Thus, log 3 1 = 0. C) lớn what exponent bởi vì we raise 3 in order to get 1/9? Since 3 -2 = 1/9, we have log 3 (1/9) = –2.

6 Example Solution Simplify: Remember that log 5 23 is the exponent lớn which 5 is raised to lớn get 23. Raising 5 to that exponent, we have It is important khổng lồ remember that a logarithm is an exponent.

7 Example Graph y = f (x) = log 3 x. Solution y 1 3 1/3 9 1/9 27 0 1 –1 2 –2 3

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

9 Solution Example Graph: y = log (x/4) – 2 in the window < 2, 8> X < 5,5>.

X < 5,5>." title="Solution Example Graph: y = log (x/4) – 2 in the window < 2, 8> X < 5,5>.">10 Equivalent Equations We use the definition of logarithm to lớn rewrite a logarithmic equation as an equivalent exponential equation or the other way around: m = log a x is equivalent to a m = x.

11 Solution Example exponential equation: a) –m = log 3 x b) 6 = log a z Rewrite each as an equivalent a) –m = log 3 x is equivalent to 3 m = x b) 6 = log a z is equivalent khổng lồ a 6 = z. The base remains the base. The logarithm is the exponent.

12 Solution Example logarithmic equation: a) 49 = 7 x b) x 2 = 9 Rewrite each as an equivalent a) 49 = 7 x is equivalent to lớn x = log 7 49 b) x 2 = 9 is equivalent khổng lồ –2 = log x 9. The base remains the base. The exponent is the logarithm.

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

14 Example Solution Solve: a) log 3 x = –3; b) log x 4 = 2. A) log 3 x = –3 x = 3 –3 = 1/27 b) log x 4 = 2 4 = x 2 x = 2 or x = –2 Because all logarithmic bases must be positive, –2 cannot be a solution. The solution is 2. The solution is 1/27. The check is left lớn the student.

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15 Solution Example Solve: a) log 6 36 = x; b) log 9 1 = t. A) log 6 36 = x 6 x = 36 x = 2 6 x = 6 2 b) log 9 1 = t 9 t = 1 9 t = 9 0 t = 0