Download presentation
Presentation is loading. Please wait.
Published byAllen Harmon Modified over 8 years ago
2
3.4 Exponential & Logarithmic Equations JMerrill, 2010
3
Properties of Logs Quick Review of 3.3
4
Rules of Logarithms If M and N are positive real numbers and b is ≠ 1: TTTThe Product Rule: llllogbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) EEEExample: log (10x) = log10 + log x YYYYou do: log7(1000x) = llllog71000 + log7x
5
Rules of Logarithms If M and N are positive real numbers and b ≠ 1: The Quotient Rule (The logarithm of a quotient is the difference of the logs) (The logarithm of a quotient is the difference of the logs) Example: You do:
6
Rules of Logarithms If M and N are positive real numbers, b ≠ 1, and p is any real number: TTTThe Power Rule: llllogbMp = p logbM (The log of a number with an exponent is the product of the exponent and the log of that number) EEEExample: log x2 = 2 log x EEEExample: log574 = 4 log57 YYYYou do: log359 CCCChallenge: = 9 log 3 5
7
Condensing Sometimes, we need to condense before we can solve: Product Rule Power Rule Quotient Rule
8
Condensing Condense:
9
Using the Rules to Condense EEEEx: YYYYou Do:
10
Bases WWWWe don’t really use other bases anymore, but since logs are often written in other bases, we must change to base 10 in order to use our calculators.
11
Change of Base Formula EEEExample log58 = TTTThis is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10! Parentheses are vital! The log key opens the ( ), you must close it!
12
3.4 Solving Exponents & Logs
13
Solving Guidelines OOOOriginalRewrittenSolution 2222x = 322x = 25x = 5 llllnx – ln3 = 0lnx = ln3x = 3 ((((1/3)x = 93-x = 32x = -2 eeeex = 7lnex = ln7x = ln7 llllogx =-110logx = 10-1x = 10-1 = 1/10 Get both parts to the same base Solve like normal Get both parts to the same base If you have a variable in the exponent position, take the log of both sides. Take the ln if you’re using e, take the log if using common logs. If you have a log on one side, exponentiate both sides
14
Solving GGGGetting all the numbers to the same base. EEEExample:
15
Solving CCCClear the exponent:
16
Solving Exponentials EEEExponentiating: eeeex = 72 llllnex = ln72 xxxx = ln72 ≈ 4.277 YYYYou should always check your answers by plugging them back in. Sometimes they don’t work because you can’t take the log of a negative number.
17
3333(2x) = 42 2222x = 14 llllog22x = log214 xxxx = log214 xxxx = log14/log2 ≈ 3.807
18
4444e2x – 3 = 2 4444e2x = 5 eeee2x = 5/4 llllne2x = ln 5/4 2222x = ln 5/4 xxxx = ½ ln 5/4 ≈ 0.112
19
2222(32t-5) – 4 = 11 2222(32t-5) = 15 ((((32t-5)= 15/2 llllog3(32t-5) = log3 15/2 2222t – 5 = log3 15/2 2222t = 5 + log3 7.5 tttt = 5/2 + ½ log3 7.5 tttt ≈ 3.417
20
eeee2x – 3ex + 2 = 0 NNNNo like terms—kinda look quadratic? ((((ex – 2)(ex – 1) = 0 SSSSet each factor = 0 and solve ((((ex – 2) = 0 eeeex = 2 llllnex = ln2 xxxx = ln2 ≈ 0.693 (e x – 1) = 0 e x = 1 lne x = ln 1 x = 0
21
Solving Logarithms EEEExponentiating with the natural log llllnx = 2 eeeelnx = e2 xxxx = e2 ≈ 7.389
22
llllog3(5x - 1) = log3(x + 7) 5555x – 1 = x + 7 4444x = 8 xxxx = 2
23
Solving Logs – Last Time 5555 + 2lnx = 4 2222lnx = -1 llllnx = - ½ eeeelnx = e - ½ xxxx = e - ½ xxxx ≈ 0.607
24
Interest Compounded Continuously IIIIf interest is compounded “all the time” (MUST use the word continuously), we use the formula where P0 is the initial principle (initial amount)
25
IIIIf you invest $1.00 at a 7% annual rate that is compounded continuously, how much will you have in 4 years? YYYYou will have a whopping $1.32 in 4 years!
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.