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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §9.5a Exponential Eqns. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 9.4. Review §. Any QUESTIONS About §9.4 → Logarithm Change-of-Base Any QUESTIONS About HomeWork §9.4 → HW-47. Summary of Log Rules.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §9.5aExponential Eqns Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 9.4 Review § • Any QUESTIONS About • §9.4 → Logarithm Change-of-Base • Any QUESTIONS About HomeWork • §9.4 → HW-47

  3. Summary of Log Rules • For any positive numbers M, N, and a with a≠ 1

  4. Typical Log-Confusion • Beware that Logs do NOT behave Algebraically. In General:

  5. Solving Exponential Equations • Equations with variables in exponents, such as 3x = 5 and 73x = 90 are called EXPONENTIAL EQUATIONS • Certain exponential equations can be solved by using the principle of exponential equality

  6. Principle of Exponential Equality • For any real number b, with b≠ −1, 0, or 1, then bx = by is equivalent to x = y • That is, Powers of the same base are equal if and only if the exponents are equal

  7. Example  Exponential Equality • Solve for x: 5x = 125 • SOLUTION • Note that 125 = 53. Thus we can write each side as a power of the same base: 5x = 53 • Since the base is the same, 5, the exponents must be equal. Thus, x must be 3. The solution is 3.

  8. Example  Exponential Equality • Solve each Exponential Equation • SOLUTION

  9. Principle of Logarithmic Equality • For any logarithmic base a, and for x, y > 0, x = y is equivalent to logax = logay • That is, two expressions are equal if and only if the logarithms of those expressions are equal

  10. Example  Logarithmic Equality • Solve for x: 3x+1 = 43 • SOLUTION 3x +1 = 43 Principle of logarithmic equality log 3x +1 = log 43 (x +1)log 3= log 43 Power rule for logs x +1= log 43/log 3 x = (log 43/log 3) – 1 • The solution is (log 43/log 3) − 1, or approximately 2.4236.

  11. Example  Logarithmic Equality • Solve for t: e1.32t = 2000 • SOLUTION e1.32t = 2000 Note that we use the natural logarithm ln e1.32t = ln 2000 Logarithmic and exponential functions are inverses of each other 1.32t = ln 2000 t = (ln 2000)/1.32

  12. To Solve an Equation of the Form at = b for t • Take the logarithm (either natural or common) of both sides. • Use the power rule for exponents so that the variable is no longer written as an exponent. • Divide both sides by the coefficient of the variable to isolate the variable. • If appropriate, use a calculator to find an approximate solution in decimal form.

  13. Example  Solve by Taking Logs • Solve each equation and approximate the results to three decimal places. • SOLUTION

  14. Example  Solve by Taking Logs • SOLUTION

  15. Example  Different Bases • Solve the equation 52x−3 = 3x+1 and approximate the answer to 3 decimals • SOLUTION Take ln of both sides

  16. Example  Eqn Quadratic in Form • Solve for x: 3x− 8∙3−x = 2. • SOLUTION • This equation is quadratic in form. • Let y = 3x then y2 = (3x)2 = 32x. Then,

  17. Example  Eqn Quadratic in Form • Solncont. • But 3x = −2 is not possible because 3x > 0 for all numbers x. So, solve 3x = 4 to find the solution

  18. Example  Eqn Quadratic in Form • Solncont.

  19. Example  Population Growth • The following table shows the approximate population and annual growth rate of the United States of America and Pakistan in 2005

  20. Example  Population Growth • Use the population model P = P0(1 + r)t and the information in the table, and assume that the growth rate for each country stays the same. • In this model, • P0 is the initial population, • r is the annual growth rate as a decimal • t is the time in years since 2005

  21. Example  Population Growth • Use P = P0(1 + r)t and the data table: • to estimate the population of each country in 2015. • If the current growth rate continues, in what year will the population of the United States be 350 million? • If the current growth rate continues, in what year will the population of Pakistan be the same as the population of the United States?

  22. Example  Population Growth • SOLUTION: Use model P = P0(1 + r)t • US population in 2005 is P0 = 295. The year 2015 is 10 years from 2005. Pakistan in 2005 is P0 = 162

  23. Example  Population Growth • SOLUTION b.: Solve for t to find when the United States population will be 350. • Some time in yr 2022 (2005 + 17.18) the USA population will be 350 Million

  24. Example  Population Growth • SOLUTION c.: Solve for t to find when the population will be the same in both countries.

  25. Example  Population Growth • Soln c.cont. • Some time year 2034 (2005 + 29.13) the two populations will be the same.

  26. WhiteBoard Work • Problems From §9.5 Exercise Set • 16, 20, 32, 34, 36, 40 • logistic difference equation by Belgian ScientistPierre Francois Verhulst

  27. All Done for Today EMP WidmarkBAC EqnCalculator

  28. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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