2. 5-4 Exponential & Logarithmic Equations Strategies and Practice

3. Objectives – Use like bases to solve exponential equations. – Use logarithms to solve exponential equations. – Use the definition of a logarithm to solve logarithmic equations. – Use the one-to-one property of logarithms to solve logarithmic equations.

4. Use like bases to solve exponential equations Equal bases must have equal exponents EX: Given 3 x-1 = 3 2x + 1 then x-1 = 2x+1  x = -2  If possible, rewrite to make bases equal EX: Given 2 -x = 4 x+1 rewrite 4 as 2 2 2 -x = 2 2x+2 then –x=2x+2  x=-2/3 Note: Isolate function if needed 3(2 x )=48  2 x =16

5. You try… 1. 4 x = 8 3 2. 5 x-2 = 25 x 3. 6(3 x+1 ) = 54 4. e –x 2 = e -3x - 4

6. Exponentials of Unequal Bases Use logarithm (inverse function) of same base on both sides of equation EX: Solve: e x = 72  lne x = ln72  xlne = ln72 x = ln72 (calc ready form) x ~ 4.277 EX: Solve: 7 x-1 = 12  log 7 7 x-1 = log 7 12 (x-1)log 7 7 = log 7 12  x-1 = log 7 12 x = 1+log 7 12 x ~ 1.277

7. You try… 1. Solve 3(2 x ) = 42 2. Solve 3 2t-5 = 15 3. Solve e 2x = 5 4. Solve e x + 5 = 60

8. Solving Logarithmic Equations Convert to exponential (inverse) form EX: Solve: lnx = -1/2 e lnx = e -1/2  x = e -1/2  x .607 EX: Solve: 2log 5 3x = 4  log 5 3x = 2 (get the log by itself) 5 2 = 3x  25/3 = x Use Properties of Logs to condense EX: Solve: log 4 x + log 4 (x-1) = ½  log 4 (x 2 -x)= ½ 4 1/2 = x 2 – x  0 = x 2 -x-2  (x-2)(x+1)  x=2

9. You try… 1. Solve lnx = -7 2. Solve 2log 5 3x = 4 3. Solve. lnx+ln(x-3) = 1 4. Solve. 5 + 2ln x = 4

10. Double-Sided Log Equations Equate powers (domain solutions only) EX: Solve: log 5 (5x-1) = log 5 (x+7) 5x – 1 = x + 7  x = 2 EX: Solve: ln(x-2) + ln(2x-3) = 2lnx Use a property: ln(x-2)(2x-3) = lnx 2 2x 2 – 7x + 6 = x 2  x 2 -7x+6=0  x = 6 & 1

11. You try… 1. Solve ln3x 2 = lnx 2. Solve log 6 (3x + 14) – log 6 5 = log 6 2x 3. Solve log 2 x+log 2 (x+5) = log 2 (x+4)

12. SUMMARY Equal bases Equal exponents Unequal bases  Apply log of given base Single side logs  Convert to exp form Double-sided logs  Equate powers Note: Any solutions that result in a log(neg) cannot be used!