Exponential+and+Logarithmic+Functions

Exponential and Logarithmic Equations
Friday, December 2, 2011 By: Mark Chubbuck
 * Transformations of Logarithmic functions:**

The logarithmic function is the inverse of the exponential funcion (Y = 2^x). Like all functions there are different coefficients that cause transformations in the function. In the case of a log function the general equation is: y = alog(k(x - d)) + c This is different from the exponential funtion in that it looks like: y = a2^(k(x-d)) + c

When regarding the coefficients in the general equations the different transformations can be seen. I both cases: a - vertical stretch - reflection in the x-axis (a<0) k - 1/k is horizontal stretch - reflection in the y-axis (k<0) d - horizontal shift c - vertical shift Using this guideline and with any combination of numbers replacing the letters, the transformations can be found and the function graphed.





**Evaluating Logarithms**
Monday December 5, 2011 Brittany Lee

Properties: a) log a 1=0 b) log a a x =x c) a^ loga^ x = x d) log a x^y = y log a x

5^x = 5^3 x = 3
 * Examples:** SOLVE FOR x ->** If there is an exponent it can ALWAYS come out in front of the log **
 * A) ** 5^x = 125

log 2^x = log 109 xlog 2 = log 109 x = log 109/lo g2 x = (approximately) 6.77
 * B)** 2^x = 109 (round to 2 decimals)

log 2^x = log 8192 2. Move x to the front xlog 2 = log 8192 3. Solve for x x = log 8192/log 2 x = 13
 * C)** 2^x = 8192 STEPS: 1. Make everything in terms of log


 * Example 2:** The doubling time for bacteria culture is 1.5 hours. The initial size of the colony is 50.
 * A)** What is the size of the colony after 4 hours
 * B)** How long will it take the colony to reach 1000


 * A)** Find time by subing the given numbers into the equation N = i(2) ^ t/d ; where N is the final amount, i is the initial size, t is the time and d is the doubling period.

N = i(2) ^ t/d = 50 (2) ^ 4/1.5 = 317.5

20 = 2 ^ t/1.5 ---> OR You can solve: N = 50(2)^x log 20 = log 2^ t/1.5 1000 = 50 (2) ^x log 20 = t/1.5log 2 20 = x^2 t/1.5 = log 20/ log 2 log 20 = log x^2 t = 6.48 log 20 = 2 log x x = log 20/ log 2 x = (apprx.) 4.32 x 1.5h x = 6.5
 * B)** 1000 = 50(2)^ t/1.5


 * Example 3 :** The richter scale magnitude R, of an earth quake is given by R = log(a/1.8) + 3.2, where a is the amplitude of the vertical ground motion in microns.
 * a)** If the amplitude of the vertical motion is 20microns what it the R value?
 * b)** What value of f will produce a 9 on the richter scale?

R = 4. 25
 * a)** R = log(a/1.8) + 3.2

5.8 = log (a/1.8) ---> WRITE IN EXPONENTIAL FORM ( y=log 10x ---> 10y = x) 10 ^ 5.8 = a/1.6 a = 1.14 x 10^6 micrometres
 * b)** 9 = log (a/1.8) + 3.2


 * HOMEWORK!! pg. 467 #11-17**

Wednesday, December7, 2011 By: Yi Lin
 * Solving Exponential Equations:**

ex.1 3^(x+2)– 3x = 72 3^x (3^2)– 3x = 72 3^x(9-1) = 72 3x = 9 X=2

ex.2 3^(2x)+ 3x = 2 (3^x )^2+ 3x – 2 = 0 (3x + 2) (3x – 1) = 0 3x = -2 No solution Or 3x = 1 X=0

Homework: p.485 #4-8, 10, 12, 18

=Logarithmic Functions Review= 1 a) 5 b) 1/4 c) 1/3 d) 3/2 e) -11/2 2. 256 3. Domain { x belongs to the real numbers / x > 0 } Range { y belongs to the real numbers } 4. 27 = x 5. log 4 29 = x 6. log ( m2 / n ) log 3 ( 4x ) 7. 2 - m 8. approximately 50.1 9. approximately 1585 10. a) (5.2, 1), (7,0), (2.5, -1) b) x = 5 c) check using graphing technology 11. a) (-3.95, -2), (-3.5, 1), ( 1, 4) b) x = -4 12. b) x=1.10 13.a) x=4 b) x = approx 0.68 14. a) log4 (x-7) + log4 2x = 2 {xer|x>7} log4 ((x-7)(2x)) = 2 log4 (2x^2-14x) = 2 2x^2 - 14x = 4^2 2x^2-14x-16 = 0 2x-16=0 x+1=0 x=8 x=-1 B) log 7 x + log 7 12 = log 7 8 log 7 (12 * x) = log 7  8 12x = 8 //x = 2/3// //c) no solution//
 * .** x=8

15. log3 4 + log3 5 = log3 (4x5) =log3 20 Therefore a + b = log3 20

16. a) A = P(1+i)^n A =500(1+0.03)^36 A = $1449

19. 2.5 = -log [H3O+] -2.5 = log [H3O+] 0.0032 = [H3O+] 0.0032/450 = 7.1 x 10^-6

pH = -log [7.1 x 10^-6] pH = 5.2

21. loga b = 1/x b= a(1/x)

logb root a = 3x2 root a =b3x^2

root a = (a1/x )3x^2

root a = a3x a = a6x loga = 6x log a loga / loga =6x 1 =6x X= 1/6