UNIT 6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Section 3 – Solving Exponential and Logarithm Functions

Properties of Logarithms 1) logbM + logbN = logb MN If 2 logs are added, they can be condensed into 1 log where the components are multiplied. 2) logbM - logbN = logb M/N If 2 logs are subtracted, they can be condensed into 1 log where the components are divided. 3) x logbM = logbMb Coefficients become exponents.

Properties of Logarithms Example 1 log x + 3 log y = log xy3 + means multiply x and y Example 2 = log xy z½ Example 3 = log x (y2z)3

Properties of Logarithms PRACTICE: supplemental packet p. 77: 11 – 40

Solving Exponential Functions Additional Properties logb1 = 0  b0 = 1 logbb = 1  b1 = b logbbx = x  bx = bx

Solving Exponential Functions Example 1 4x = 23 We solve by using the inverse function The inverse of an exponential is a logarithm 4x = 23 *Take the log4 of both sides log4 (4)x = log4 (23) *log4(4) = 1 leaving just x x = log423 = 2.2618

Solving Exponential Functions Example 2 62x - 8 + 4 = 15 * Isolate the exponential 62x - 8 = 11 *Take the log6 of both sides log6 (6)2x-8 = log6 (11) 2x – 8 = log611 2x – 8 = 1.3383 2x = 9.3383 x = 4.6691

Solving Exponential Functions PRACTICE: supplemental packet p. 78: 11 – 18,

Solving Exponential Functions Example 1 4 log (4x – 2) = 20 * Isolate the log log (4x – 2) = 5 *Exponentiate both sides 10log(4x-2) = 105 raise 10 to each side 4x – 2 = 105 * base 10 cancels the log 4x – 2 = 100000 x = 25000.5

Solving Exponential Functions Example 2 log5(3x) – log5(4x-2) = 3 log5(3x/4x-2) = 3 *Condense using properties. *Exponentiate both sides 3x = 53 4x – 2 3x = 125(4x – 2) * Multiply by 4x - 2 3x = 500x – 250 * Distribute -497x = -250 x = .503

Solving Exponential Functions Example 3: log6 (x) + log6(x+5)= 2 log6 (x)(x + 5) = 2 * Product property log6 (x2 + 5x) = 2 * Dist. property x2 + 5x = 62 * Exponentiation x2 + 5x – 36 = 0 *Solve by factoring (x + 9)( x – 4) = 0 x = -9, 4 (only 4 works) PRACTICE: packet p. 78: 19 – 49

Solving Natural Log e = 2.71828 The inverse to y = ex is y = ln(x) (or the natural log of x) y = ex is the same as y = (2.71828)x y = ln(x) is the same as y = loge(x)

Solving Natural Log Example 1 3 ln(2x + 1) = 24 *Isolate ln(2x + 1) = 8 *Exponentiate both sides eln(2x + 1) = e8 *Use shift, ln for e^ 2x + 1 = 2980.96 x = 1489.98

PRACTICE Solve: ln(5x + 3) = 5 eln(5x+3) = e5 5x + 3 = 148.41

Example 2 Example 2: Solve e4x-1 + 4 = 8 *Isolate e4x-1 = 4 *Take ln of both sides ln(e4x-1) = ln(4) 4x – 1 = 1.3863 x = .5966

Practice Solve: 3ex+4 = 81 ex+4 = 27 ln(ex+4) = ln(27) x + 4 = 3.2958

Example 3 2 ln (x) – ln(7)= 7 ln(x2) – ln(7)= 7 * Power property ln (x2/7) = 7 * Quotient property x2/7 = e7 * Exponentiation x2/7 = 1096.63 x2 = 7676.43 x = 87.62

PRACTICE Supplemental Packet p. 79

Exponential Applications y = a(b)x Example 1 A) Function y = 12(1.07)x B) y = 12(1.07)30 = 91,347 people C) 19 = 12(1.07)x *y=19 for 19,000 people 1.583 = 1.07x *Divide by 12 log1.071.583 = log1.071.07x 6.8 = x During 1996

Exponential Applications y = a(b)x Example 2 A) Function y = 35(.91)x B) y = 35(.91)3 = $26,375 C) 5 = 35(.91)x 1/7 = .91x *Divide by 35 log.91 (1/7) = log.91 .91x 20.633 = x After 20.6 years, it will be worth $5000

Exponential Applications y = a(b)x Example 3 A) Function y = 20(e).06x B) y = 20(e).06(15) = $49,192 C) 80 = 20(e).06x 4 = e.06x *Divide by 20 ln (4) = ln(e.06x) 1.3863 = .06x x = 23.1 years

Exponential Applications y = a(b)x PRACTICE Supplemental packet p. 80: 1 – 4 and p. 81

Applications from data Example 1 What is the starting amount and growth factor? A) Function y = 3(2)x B) y = 3(2)20 = 3,145,728 bacteria C) 1,000,000 = 3(2)x 333,333.3 = 2x (Divide by 3) log2333,333.3 = log22x x = 18.35 minutes

Exponential y = a(b)x rate = (End – Start)/Start value Example 2 A) rate = (27200 – 32000)/32000 = -.15 Function y = 32000(.85)x B) y = 32000(.85)7 = $10258.47 C) 6000 = 32000(.85)7 .1875 = .85x (Divide by 32000) log.85.1875 = log.85.85x x = 10.3 years

Exponential y = a(b)x Example 3 PRIZM: Enter years as list 1 and population as list 2 Press F2 (CALC), F3 (REG), F6 F2(EXP), F2(abx). This is your function. y = 26.31 (1.0836)x

Exponential y = a(b)x Example 3 Press F6 and EXE to copy this into your Graph/Table app. B) Make your prediction in the TABLE app C) Solve 300 = 26.31 (1.0836)x Enter 300 as Y2 and graph both. F5 (G-SOLV) and F5 (INTSECT)

PRACTICE Complete p. 82 and 83