2. Solving Exponential and Logarithmic Equations JMerrill, 2005 Revised, 2008 LWebb, Rev. 2014

3. Same Base SSSSolve: 4x-2 = 64x  4 x-2 = (43)x x-2 = 43x  x –2 = 3x  - 2 = 2x 1 = x If b M = b N, then M = N 64 = 4 3 If the bases are already =, just solve the exponents

4. You Do  Solve 27 x+3 = 9 x-1

5. Review – Change Logs to Exponents llllog3x = 2 llllogx16 = 2 llllog 1000 = x 3 2 = x, x = 9 x 2 = 16, x = 4 10 x = 1000, x = 3

6. Using Properties to Solve Logarithmic Equations 1111.Condense both sides first (if necessary). 2222a.If there is only one side with a logarithm, then rewrite the equation in exponential form. 2222b.If the bases are the same on both sides, you can cancel the logs on both sides. 3333.Solve the simple equation.

7. Solving by Rewriting as an Exponential SSSSolve log4(x+3) = 2 44442 = x+3 11116 = x+3 11113 = x

8. You Do SSSSolve 3ln(2x) = 12 lllln(2x) = 4 RRRRealize that our base is e, so eeee4 = 2x xxxx ≈ 27.299 YYYYou always need to check your answers because sometimes they don’t work!

9. Example: Solve for x llllog36 = log33 + log3xproblem llllog36 = log33xcondense  6 = 3xdrop logs  2 = xsolution

10. You Do: Solve for x llllog 16 = x log 2problem llllog 16 = log 2xcondense  1 6 = 2xdrop logs  x = 4solution

11. Example  7xlog 2 5 = 3xlog 2 5 + ½ log 2 25  log 2 5 7x = log 2 5 3x + log 2 25 ½  log 2 5 7x = log 2 5 3x + log 2 5 1  log 2 5 7x = log 2 (5 3x )(5 1 )  log 2 5 7x = log 2 5 3x+1  7x = 3x + 1  4x = 1 

12. You Do: Solve for x  l og4x = log44problem  = log44condense 4drop logs ccccube each side  X = 64solution

13. You Do  Solve:log 7 7 + log 7 2 = log 7 x + log 7 (5x – 3)

14. You Do Answer  Solve:log 7 7 + log 7 2 = log 7 x + log 7 (5x – 3)  log 7 14 = log 7 x(5x – 3)  14 = 5x 2 -3x  0 = 5x 2 – 3x – 14  0 = (5x + 7)(x – 2)  Do both answers work? NO!!

15. Using Properties to Solve Logarithmic Equations IIIIf the exponent is a variable, then take the natural log of both sides of the equation and use the appropriate property. TTTThen solve for the variable.

16. Example: Solving  2 x = 7 problem lllln2x = ln7 take ln both sides xxxxln2 = ln7power rule  x = divide to solve for x = 2.807

17. Example: Solving  e x = 72problem  l nex = ln 72take ln both sides  x lne = ln 72power rule = 4.277solution: because ln e = ?

18. You Do: Solving  2 ex + 8 = 20problem ex = 12subtract 8  e x = 6divide by 2  l n ex = ln 6take ln both sides  x lne = 1.792power rule x = 1.792(remember: lne = 1)

19. Example SSSSolve 5x-2 = 42x+3 lllln5x-2 = ln42x+3  use log rules ((((x-2)ln5 = (2x+3)ln4  distribute xxxxln5 – 2ln5 = 2xln4 + 3ln4  group x’s xxxxln5 – 2x ln4 = 3ln4 + 2ln5  factor GCF xxxx(ln5 – 2ln4) = 3ln4 + 2ln5  divide xxxx = 3ln4 + 2ln5 ln5 – 2ln4 xxxx≈-6.343

20. You Do:  Solve 3 5x-2 = 8 x+3  ln 3 5x-2 = ln 8 x+3  (5x-2)ln3 = (x+3)ln8  5xln3 – 2ln3 = xln8 + 3ln8  5xln3 – xln8 = 3ln8 + 2ln3  x(5ln3 – ln8) = 3ln8 + 2ln3  x = 3ln8 + 2ln3 5ln3 – ln8 5ln3 – ln8 x = 2.471 x = 2.471

21. Final Example  How long will it take for $25,000 to grow to $500,000 at 9% annual interest compounded monthly?

22. Example