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

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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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Chabot Mathematics

§9.5aExponential Eqns

Bruce Mayer, PE

MTH 55

9.4

Review §
• §9.4 → Logarithm Change-of-Base
• §9.4 → HW-47
Summary of Log Rules
• For any positive numbers M, N, and a with a≠ 1
Typical Log-Confusion
• Beware that Logs do NOT behave Algebraically. In General:
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
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
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.
Example  Exponential Equality
• Solve each Exponential Equation
• SOLUTION
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
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.
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

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.
Example  Solve by Taking Logs
• Solve each equation and approximate the results to three decimal places.
• SOLUTION
Example  Different Bases
• Solve the equation 52x−3 = 3x+1 and approximate the answer to 3 decimals
• SOLUTION

Take ln of both sides

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

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
Example  Population Growth
• SOLUTION c.: Solve for t to find when the population will be the same in both countries.
Example  Population Growth
• Soln c.cont.
• Some time year 2034 (2005 + 29.13) the two populations will be the same.
WhiteBoard Work
• Problems From §9.5 Exercise Set
• 16, 20, 32, 34, 36, 40
• logistic difference equation by Belgian ScientistPierre Francois Verhulst
All Done for Today

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Appendix

Bruce Mayer, PE