Mini Lecture 12.1 Exponential Functions. e d. 2. Graph each function by making a table of coordinates.

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1 Learning Ojectives:. Evaluate eponential functions.. Graph eponential functions.. Evaluate functions with ase e.. Use compound interest formulas. Mini Lecture. Eponential Functions Eamples:. Approimate each numer using a calculator, round your answer to two decimal places. a... c.. Graph each function y making a tale of coordinates. a. ( ). f g( ). e d. 0.5 e. Graph f and g in the same rectangular coordinate system. Select integers from to for. Then descrie how the graph of g is related to the graph of f. a. f ( ) and g( ). f ( ) and g( ). Find an accumulated value of an investment of $7000 for 5 years at an interest rate of % if the money is: a. compounded monthly.. compounded semi-annually. c. compounded continuously. Use the compound interest formulas to solve. Round answers to the nearest cent. The eponential function f with ase is f ( ) or y, > 0 and and is any real numer. The domain of an eponential function consists of all real numers. The range of an eponential function consists of all positive real numers. If >, the graph of the eponential function will go up to the right and is an increasing function. The larger the value of, the steeper the increase. If 0 < <, the graph of the eponential function goes down to the right and is a decreasing function. The small the value of, the steeper the decrease. An irrational numer, e, often appears as a ase in applied eponential functions. The numer e is called the natural ase. " e " Copyright 0 Pearson Education, Inc. ML-8

2 The function f ( ) e is called the natural eponential function. Formulas for compound interest: r nt * for n compounding per year A P( ) n rt * for continuous compounding A Pe After t years, the alance A in an account with principal P and as annual interest rate r. Answers:. a c..00 d a... a. The graph of g is the graph of f shifted unit to the left. The graph of g is the graph of f shifted up units. a. $ $85.96 c. $859.8 Copyright 0 Pearson Education, Inc. ML-8

3 Mini Lecture. Logarithmic Functions Learning Ojectives:. Change from logarithmic to eponential form.. Change from eponential to logarithmic form.. Evaluate logarithms.. Use asic logarithmic properties. 5. Graph logarithmic functions. 6. Find the domain of a logarithmic function. 7. Use common logarithms. 8. Use natural logarithms. Eamples:. Write each equation in its equivalent eponential form. a. 5 log. log 6 c. log 5 8 y. Write each equation in its equivalent logarithmic form. a. = 8. = 8 c. e y =. Evaluate. a. log log 5 5 c. log 5 5 d. log e. log 7 f. log log5 g. 5 h. ln i. ln e. Graph f ( ) and g( ) log in the same rectangular coordinate system. 5. Find the domain of each function: a. f ( ) log ( ). g( ) ln(5 ) The function f ( ) log is the logarithmic function with ase. The logarithmic form y log is equivalent to the eponential form y. The domain of a logarithmic function of the form f ( ) log is the set of all positive real numers. The domain of f ( ) log [ g( )] consists of all for which g() > 0. The logarithmic function with ase 0 is called the common logarithmic function. Properties of Common Logarithms General Properties Common Logarithms Natural Logarithms log 0 log ln = 0 0 log log0 ln e = log log 0 ln e = log log 0 e ln = Copyright 0 Pearson Education, Inc. ML-8

4 Answers:. a c. 5 y 8. a. log8. log8 c. y log e. a.. c. d. e. 0 f. g. h. 0 i.. 5. a. (, ). (, 5) Copyright 0 Pearson Education, Inc. ML-85

5 Learning Ojectives:. Use the product rule.. Use the quotient rule.. Use the power rule.. Epand logarithmic epressions. 5. Condense logarithmic epressions. 6. Use the change-of-ase property. Mini Lecture. Properties of Logarithms Eamples: Epand.. a. log ( 8). log 8 ( 8) c. log 5 (5). a. log. e ln 5 0 c. log. a. log 7. y log z c. log 5. a. log y. log5 y 6y c. log 7 Condense. Write as a single logarithm. 5. a. log8 5 log 8. log 6 8 log 6 log 6 9 c. log 5 log d. log log ( ) log Use the change-of-ase property and your calculator to find the decimal approimation. 6. a. log 6. log 9 7 c. log 7 5 Students must know properties and rules of logarithms and since this will e new to most students, a lot of practice is recommended. As rules are introduced, show several eamples with numers instead of letters. Make sure students know how to use their calculators to find decimal approimations. Answers:. a. log log 8. log 8 c. log 5. a. log. ln 5 c. log. a. log 7. log y log z c. log5. a. log log y. log 5 log5 y c. log7 6 log7 y log7 5. a. log 5 8. log 6 c. log 50 d. log 6. a c..97 Copyright 0 Pearson Education, Inc. ML-86

6 Mini Lecture. Eponential and Logarithmic Equations Learning Ojectives:. Use like ases to solve eponential equations.. Use logarithms to solve eponential equations.. Use eponential form to solve logarithmic equations.. Use the one-to-one property of logarithms to solve logarithmic equations. 5. Solve applied prolems involving eponential and logarithmic equations. Eamples:. Solve a c Find the solution set and then use a calculator to otain a decimal approimation to two decimal places for the solution. a Solve. a. log ( ). log log ( ) c. log ( ) log ( ) d. 5ln 5 An eponential equation is an equation containing a variale with an eponent. To solve eponential equations, epress each side of the equal as a power of the same m n ase and then set the eponents equal to each other. If, then m n. When using rational logarithms to solve eponential equations, first, isolate the eponential epressions. Net, take the natural logarithm on oth sides of the equation, simplify and solve for the variale. A logarithmic equation is an equation containing a variale in a logarithmic epression. ALWAYS check proposed solutions of a logarithmic equation in the original equation. Eclude from the solution set any proposed solution that produces the logarithm of a negative numer or the logarithm of 0. Answers:. a.. d. e ln0 ln 6 c.. a.. 6. ln a. 9. c. ln Copyright 0 Pearson Education, Inc. ML-87