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Solving Exponential and Logarithmic Equations

Learn to solve exponential and logarithmic equations using properties of logarithms. Explore real-world examples modeling population and decline. Includes symbolic and graphical solutions.

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Solving Exponential and Logarithmic Equations

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  1. Chapter 5 Exponential and Logarithmic Functions

  2. Exponential and Logarithmic Equations 5.6 Solve exponential equations Solve logarithmic equations

  3. Exponential Equation An equation in which one or more variables occur in the exponent of an expression iscalled an exponential equation. We use Property 4 of logarithms,given by loga (mr )=r logam,to solve exponential equations.

  4. World population in billions during year x can be modeled by P(x) = 7(1.01)x – 2011. Solve Example: Modeling world population the equation 7(1.01)x – 2011 = 8 symbolically to predict the year when world population reached 8 billion.

  5. Solution Example: Modeling world population

  6. This model predicts that world population might reach 8 billion during 2024. Example: Modeling world population

  7. Bluefin tuna are large fish that can weigh 1500 pounds and swim at speeds of 55 miles per hour. Because they are used for sushi, a prime fish can be worth over $50,000. As a result, the western Atlantic bluefin tuna have had their numbers decline exponentially. Their numbers in thousands from 1974 to 1991 can be modeled by the formula f(x) = 230(0.881)x, where x is years after 1974. (In more recent years, controls have helped to slow this decline. (Source: B. Freedman, Environmental Ecology.) Example: Modeling the decline of bluefin tuna

  8. (a)Estimate the number of bluefin tuna in 1974 and 1991. (b)Determine symbolically the year when they numbered 50 thousand. Example: Modeling the decline of bluefin tuna

  9. Solution (a) To determine their numbers in 1974 and 1991, evaluate f(0) and f(17). f(0) = 230(0.881)0 = 230(1) = 230 f(17) = 230(0.881)17  26.7 Bluefin tuna decreased from 230 thousand in 1974 to fewer than 27 thousand in 1991. Example: Modeling the decline of bluefin tuna

  10. (b) Solve the equation f(x) = 50 for x. Example: Modeling the decline of bluefin tuna

  11. They numbered about 50 thousand in 1974 + 2.04  1986. Example: Modeling the decline of bluefin tuna

  12. Solve each equation. Example: Solving exponential equations symbolically

  13. Example: Solving exponential equations symbolically

  14. Example: Solving exponential equations symbolically

  15. Example: Solving exponential equations symbolically

  16. Solve e–x + 2x = 3 graphically. Approximate all solutions to the nearest thousandth. Solution The graphs of Y1 = e^(–X) + 2X and Y2 = 3 intersect near the points (–1.92, 3) and … Example: Solving exponential equations symbolically

  17. (1.37, 3). The solutions are approximately –1.92 and 1.37. Example: Solving exponential equations symbolically

  18. Logarithmic Equations Logarithmic equations contain logarithms. Like exponential equations, logarithmic equations also occur in applications. To solve a logarithmic equation, we use the inverse property

  19. Solve Solution Example: Solving a logarithmic equation

  20. In developing countries, there is a relationship between the amount of land a person ownsand the average daily calories consumed. This relationship is modeled by the formulaC(x) = 280ln(x + 1) + 1925, where xis the amount of land owned in acres and 0 ≤ x ≤ 4. (a)Find the average caloric intake for a person who owns no land. Example: Solving a logarithmic equation symbolically

  21. (b)A graph of Cis shown.Interpretthe graph. (c)Determine symbolically the number of acres owned by someone whose average intakeis 2000 calories per day. Example: Solving a logarithmic equation symbolically

  22. (a) Since C(0)= 280ln(0 + 1)+ 1925 = 1925, a person without land consumes anaverage of 1925 calories per day. Example: Solving a logarithmic equation symbolically

  23. (b) As the amount of land x increases, the caloric intake y also increases. Example: Solving a logarithmic equation symbolically However, the rate of increase slows. This would be expected because there is a limit to the number of calories an average person would eat, regardless of his or her economic status.

  24. (c) Solve the equation C(x) = 2000. A person who owns about 0.3 acre has an average intake of 2000 calories per day. Example: Solving a logarithmic equation symbolically

  25. Solve each equation. Example: Solving logarithmic equations symbolically

  26. Solution Example: Solving logarithmic equations symbolically

  27. –1 is not a solution since log2 (–1) is undefined. The only solution is 1. Example: Solving logarithmic equations symbolically

  28. Substituting x= 0and x= –4 in the given equation shows that 0 is a solution but–4is not a solution. Example: Solving logarithmic equations symbolically

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