MATH 234, Calculus for BusinessLecture 3, Textbook Sections 2.4, 2.5Exponential and Logarithmic FunctionsJanuary 25th, 2017Announcements.(1) In Math 234 site, go to the side link “Frequently Asked Questions”. (2) HW1 is due Thursday 1/26,11:59pm; late HWs are not accepted.Section 2.4, Exponential FunctionsDefinition. The exponential function with base a iswhere and .Theorem 1 (Properties of Exponents; see Section R.6). Fix a, b. For all real numbers x and u,(1) (ax)u=(2) (ax)(au) =(3)axau=(4) (ab)x=(5) a0=(6)abx=(7) If a 6= 1 and ax= au,then .WARNING! (A common mistake to avoid). (a+b)xax+bx. For example, (a+b)2= .Remark. Note that 1x= for all x. Also, .Example 1 (Integer Powers). For positive integers n,• 74=• 7n=• 7−n=• (72)3=• 72∗ 73=•753=Also, 70=1Example 2 (Fractional Powers, Part 1). Observe that 64 = 641= . So641/3=Definition. Let a > 0 and let n be a positive integer, then is the n-th root of a.Remark. An alternative notation is .Example 3 (Fractional Powers, Part 2).645/3=Remark. Since√−5 is not a real number, (−5)xis not defined for .Theorem 2 (Properties of Graphs of Exponential Functions). Fix . For the graph of f(x) = ax,(1) Domain is . Range is .(2) The line is .(3) The is always .(4) The graph can take 2 shapes: when a > 1, then graph ;when 0 < a < 1, then graph .Graph of f(x) = 2xis Graph of f(x) = (1/2)xisExample 4. Solve 3x22x= 144Solution.2Simple InterestDefinition. The initial investment (or principal) is denoted .The interest rate is denoted .Remark. The interest rate of is written .Definition. Let t = time in years. For simple interest, the interest earned isI(t) =Example 5. Alice deposits $10,000 to an account with simple interest of 5 percent. After 10 years, the interestearned isI(10) =So, Alice’s account has after 10 years.Compound InterestDefinition (Compound Interest). Recall, principal is P dollars and interest rate is r%. Compound interestcomputes interest using principal P and previously accumulated interest. If an account compounds m timesper year, then the compounded amount, A(t), after t years isA(t) =Remark. Derivation of this formula is in Section 2.4, discussion before Example 4.Remark. • Compound interest than simple interest.• Larger principal means .• Bigger interest rates also means .Example 6 (Faster or Slower Compounding?). Alice has $10,000 and wants to open a savings account. BothBank A and Bank B pay 5% interest, but Bank A compounds quarterly while Bank B compounds every day.Which bank should she choose?Solution. Step 1: Find Formula for Compound Amount for Both Banks. The principal is.For Bank A, , so the compound amount for the savings account in Bank A isA(t) =For Bank B, , so the compound amount for the savings account in Bank B isB(t) =3Example 7 (Faster or Slower Compounding?). Alice has $10,000 and wants to open a savings account. BothBank A and Bank B pay 5% interest, but Bank A compounds quarterly while Bank B compounds every day.Which bank should she choose?Solution. Step 2: Compare A(t) vs B(t)Savings Account 1 year 5 years 10 years 20 years 40 yearsBank A, A(t) $10,509 $12,820 $16,436 $27,014 $72,980Bank B, B(t) $10,512 $12,840 $16,486 $27,181 $73,880Alice should choose ,Faster compounding gives .What is the fastest compounding? .Continuous Compound InterestRemark. The function f(m) =1 +1mmhas :The isDefinition. Recall, principal is P dollars and interest rate is r percent. The continuously compoundedamount, A(t), after t years isA(t) =Remark. Derivation of this formula is in Section 2.4, discussion before Example 5.Remark. In terms of growth rate:< <Example 8. Bob needs to borrow a student loan of $10,000. Lenders A, B, and C all charge 5% interest.However, Lender A uses the simple interest formula, Lender B uses compound interest and compounds daily(m = 365), while Lender C uses continuous compound interest.Which lender should Bob choose?Answer: Bob should choose beacuse4Section 2.5, Logarithmic FunctionsDefinition. Fix f(x). The inverse function, g(x), isif , thenNamely, g(x)Example 9. Find the inverse of f(x) = 3(x − 4)2+ 5.Solution. Step 1: Write y = f(x), and switch x ←→ y.Solution. Step 2: Solve for y, and let g(x) = y.The inverse function is g(x) = .For the above example,f(4) = g(5) =f(5) = g(8) =f(6) = g(17) =Graphing both f(x) and g(x) on the same graph looks likeThe graph of g(x) is the graph of f(x) .5Definition. The logarithm function with base a, writtenis defined as . Again, .Example 10. Let a = 2, then23= log2(8) =210= log2(1024) =Remark. The value of y = loga(x) is .Example 11. Compute log3(81).Example 12. Graphing y = 2xand y = log2(x) on the same graph looks likeThe graphs are .Theorem 3 (Properties of Graphs of Logarithmic Functions). Fix .For the graph of f(x) = loga(x),(1) Range is . Domain is .(2) The lineis .(3) The is always .(4) The graph can take 2 shapes: when a > 1, then graph ;when 0 < a < 1, then graph .6Graph of f(x) = log2(x) is Graph of f(x) = log1/2(x) isRemark. log10(x) is commonly written as .loge(x) is commonly written as .Theorem 4 (Properties of Logarithms). Fix a, b. For all real numbers x, u, and r(1) loga(1) =(2) loga(a) =(3) loga(x ∗ u) =(4) loga(x/u) =(5) loga(xr) =(6) loga(ar) =(7) aloga(x)=(7) loga(x) =(8) loga(x) =WARNING! (Common mistakes to avoid). First,loga(x + u) loga(x) + loga(u)Also,loga(x)rr ∗ loga(x)Finally,logb(x)logb(a)xaandln(x)ln(a)xa7Example 13. Simplify log5(x3p1 − y2) using algebra and the properties of logarithms.Solution.Example 14 (Doubling time). Alice deposits $10,000 to a saving account. This account pays 5% interest andcompounds continuously.How long will it take for the account to have $20,000 (twice the original)?Solution. Step 1: Determine the function for compounded amountThe principal is and the interest rate is .Since the account compounds continuously, the compound amount function isA(t) =Solution Step 2: Solve for t.Alice’s savings account will double in