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Understanding Logarithmic Functions for Effective Math Problem Solving

Learn about logarithmic functions and how they relate to exponential functions, domain considerations, common logarithms, natural logarithms, graphing techniques, and solving equations using logarithms. Master these concepts to excel in math problem-solving.

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Understanding Logarithmic Functions for Effective Math Problem Solving

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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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