7 General Form of Exponential Function y = b x where b > 1 Domain: All realsRange:y > 0x-intercept: Noney-intercept: (0, 1)
8 General Form of Exponential Function y = b (x + h) + k where b > 1 h moves graph left or right (opposite way)k move graph up or down (expected way)So y=3(x+2) + 3 moves the graph 2 units to the left and 3 units up(0, 1) to (– 2, 4)
9 Graph:XY-2-11239311/30.1110.037Decreasing for all of x
10 Graph:XY-2-11230.1250.250.5124Increasing for all of x
11 Graph:XY-2-1123-0.5-1-2-4-8-16Decreasing for all of x
12 Graph:XY-2-112384210.50.25Decreasing for all of x
17 Properties of Logarithms (Shortcuts) logb1 = 0 (because b0 = 1)logbb = 1 (because b1 = b)logb(br) = r (because br = br)blog b M = M (because logbM = logbM)
18 Properties of Logarithms logb(MN)= logbM + logbNEx: log4(15)= log45 + log43logb(M/N)= logbM – logbNEx: log3(50/2)= log350 – log32logbMr = r logbMEx: log7 (103) = 3 log7 10If logbM = logbN Then M = Nlog11 (1/8) = log11 8-1
19 TRY THESE TO SOLVE THESE: First I would change to exponential form!!
20 It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution.
21 Can anyone give us an explanation ? Our final concern then is to determine why logarithms like the one below are undefined.Can anyone give us an explanation ?One easy explanation is to simply rewrite this logarithm in exponential form.We’ll then see why a negative value is not permitted.What power of 2 would gives us -8 ?Hence expressions of this type are undefined.
25 Review – Change Logs to Exponents 32 = x, x = 9log3x = 2logx16 = 2log 1000 = xx2 = 16, x = 410x = 1000, x = 3
26 Example 7xlog25 = 3xlog25 + ½ log225 log257x = log253x + log225 ½ 7x = 3x + 14x = 1
27 Example 1 – Solving Simple Equations a. 2x = 32 2x = 25 x = 5b. ln x – ln 3 = 0 ln x = ln 3 x = 3c = 9 3– x = 32 x = –2d. ex = 7 ln ex = ln 7 x = ln 7e. ln x = –3 eln x = e– 3 x = e–3f. log x = –1 10log x = 10–1 x = 10–1 =g. log3 x = 4 3log3 x = 34 x = 81
28 Example 6 – Solving Logarithmic Equations a. ln x = 2eln x = e2x = e2b. log3(5x – 1) = log3(x + 7)5x – 1 = x + 74x = 8x = 2
30 Given the original principal, the annual interest rate, and the amount of time for each investment, and the type of compounded interest, find the amount at the end of the investment.1.) P = $1,250; r = 8.5%; t = 3 years; quarterly2.) P = $2,575; r = 6.25%; t = 5 years, 3 months; continuously