1. Logarithms and Exponential Equations Ashley Berens Madison Vaughn Jesse Walker

2. Logarithms Definition- The exponent of the power to which a base number must be raised to equal a given number.

3. Evaluating logarithms If b > 0, b ≠ 1, and x >0 then… Logarithmic Form …. Exponential Form log x = y b = x by Examples… – log 81 = x 3 = 81 x = 4 3 x – 2 = 2 x = 1 x

4. Basic Properties logь1=0 logьb=1 logьb =x ь b =x, x>0 x Log x ]- Inverse properties

5. Examples of Basic Properties Log 125 = 5 = 125 x = 3 log 81= 9 = 81 x = 2 5 x 9 x 12 12 ( 12’s Cancel) Log = 4.7 3 3 1 ( 3’s Cancel) Log = 1 log 4.7 log 4.7 / / log / /

6. Common Logarithms If x is a real number then the following is true… Log 1 = 0 Log 10 = 1 Log 10 = x 10 = x, x > 0 x logx ]- Inverse Properties

7. Common Logs Log 0.001 log = log -3 = log 10 log = -3 Log(-5) 10 = -5 NO SOLUTION ( Because it’s a negative) 1/ 10001/103 -3 Log -0 10 = 0 NO SOLUTION Log 10,000 10 = 10,000 x = 4 x x x

8. Natural logs If x is a real number then…. ln 1 = 0 ln e = 1 ln e = x e = x, x > 0 x ln x ]- Inverse properties

9. Natural log examples ln e ln = 0.73 ln ( -5) No Solution ( Cant have a natural long of a negative) 0.73 ln 32 e = 32 x = (Use Calculator) e e = 6 x ln 6

10. Expanding Logarithms log12x y = log12 logx + log y = log12 + 5logx – 2log y ln = lnx - ln = 2lnx – ½ ln (4x+1) 5-2 5 X 2 √ 4x+1 2

11. Condensing logarithms -5 log (x+1) + 3 log (6x) = 3log (6x) – 5log (x+1) = log 6x - log (x+1)5 = log 2 2 22 22 2 (6a) 3 (x+1) 2

12. Change of base log 5 = log5 log3 (Use Calculator) =1.34649… log 6 = log6 log ½ (Use Calculator) = -2.5849… log 4212 = log 4212 log 78 = (Use Calculator) = 1.9155… log 33 = log 33 log 15 = (Use Calculator) = 1.2911… 3 ½ For any positive real numbers a, b and x, a ≠1, b ≠1 78 15

13. Exponential Functions  Exponential functions are of the form f(x)=ab, where a≠0, b is positive and b≠1. For natural base exponential functions, the base is the constant e.  If a principle P is invested at an annual rate r (in decimal from), then the balance A in the account after t years is given by:  x

14. Formulas A = P( 1+r/n ) When compounded n times in a year. A = Pe When compounded continuously. nt rt

15. Exponential Examples… New York has a population of approximately 110 million. Is New York's population continues at the described rate, predict the population of New York in 10… –A. 1.42% annually F(x) = 110 * (1+.0142) F(x)= 110 * 1.0142 F(10) = 126,657,000 –B. 1.42% Continuously N = Pe N(t) = 110e N(t) = 126,783,000 t t rt (.0142 * t)

16. Finding growth and decay 562.23 * 1.0236 t If the number is more than one than it is an exponential increase. If it is less than one than it is a exponential decrease. <- Exponential Growth