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SECTION 4.4

SECTION 4.4. LOGARITHMIC FUNCTIONS. LOGARITHMIC FUNCTIONS. The logarithm (base b) of a number is the power to which b must be raised to get that number. EXAMPLES:. (a) log 4 16 =. 2. (e) log 5 1 =. 0. (b) log 3 27 =. 3. (f) log 2 1/8 =. - 3. (g) log 10 .1 =. - 1.

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SECTION 4.4

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  1. SECTION 4.4 • LOGARITHMIC FUNCTIONS

  2. LOGARITHMIC FUNCTIONS The logarithm (base b) of a number is the power to which b must be raised to get that number.

  3. EXAMPLES: (a) log 4 16 = 2 (e) log 5 1 = 0 (b) log 3 27 = 3 (f) log 2 1/8 = - 3 (g) log10 .1 = - 1 (c) log e e 4 = 4 (d) log 2 32 = 5 (h) log 9 27 = 3/2

  4. LOGARITHMIC FUNCTIONS Recall that only one-to-one functions have an inverse. Exponential functions are one-to-one. Their inverses are logarithmic functions.

  5. LOGARITHMIC FUNCTIONS Example: Change the exponential expressions to logarithmic expressions. 1.23 = m eb = 9 a4 = 24

  6. LOGARITHMIC FUNCTIONS Example: Change the logarithmic expressions to exponential expressions. loga4 = 5 loge b = - 3 log3 5 = c

  7. DOMAIN OF A LOGARITHMIC FUNCTION Since the logarithmic function is the inverse of the exponential, the domain of a logarithmic is the same as the range of the exponential.

  8. DOMAIN OF A LOGARITHMIC FUNCTION Example: Find the domain of the functions below: F(x) = log2 (1 - x)

  9. SPECIAL LOGARITHMS Logarithm to the base 10. Ex: log 100 = 2 COMMON LOGARITHM Logarithm to the base e. Ex: ln e 2 = 2 NATURAL LOGARITHM

  10. THE NATURAL LOGARITHMIC FUNCTION • Graph the function g(x) = lnx in the same coordinate plane with f(x) = ex • Notice the symmetry with respect to the line y = x.

  11. f(x) = ex g(x) = lnx Compose the two functions: g(f(x)) = ln ex = x f(g(x)) = eln x = x We can see graphically as well as algebraically that these two functions are inverses of each other.

  12. Given f(x) = bx Then f -1(x) = log b x

  13. GRAPHS OF LOGARMITHMIC FUNCTIONS 1. The x-intercept is 1. 2. The y-axis is a vertical asymptote of the graph. 3. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1. 4. The graph is continuous.

  14. GRAPHING LOGARITHMIC FUNCTIONS USING TRANSFORMATIONS Graph f(x) = 3log(x – 1). Determine the domain, range, and vertical asymptote of f.

  15. EXAMPLE Graph the function f(x) = ln(1 - x). Determine the domain, range, and vertical asymptote.

  16. SOLVING A LOGARITHMIC EQUATION Solve: log3(4x – 7) = 2 Solve: logx64 = 2

  17. USING LOGARITHMS TO SOLVE EXPONENTIAL EQUATIONS Solve: e2x = 5

  18. EXAMPLE DO EXAMPLE 10 ON ALCOHOL AND DRIVING

  19. CONCLUSION OF SECTION 4.4

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