1. PRECALCULUS I EXPONENTIAL & LOG EQUATIONS Dr. Claude S. Moore Danville Community College

2. EXPONENTIAL & LOG INVERSE PROPERTIES EXPONENTIAL & LOG INVERSE PROPERTIES 1. log a a x = x ln e x = x. 2. a log a x = x e ln x = x.

3. ONE-TO-ONE PROPERTIES 1. log a x = log a y iff x = y. 2. a x = a y iff x = y. 3. a x = b x iff a = b.

4. TO SOLVE... Exponential equation: Isolate exponential expression; take log of both sides and solve. Logarithm equation: Rewrite as exponential equation and solve.

5. EXAMPLE 1 Solve (no calculator): 3 x-1 = 243 3 x-1 = 3 5 Thus, x - 1 = 5 or x = 6.

6. EXAMPLE 2 Solve (3-decimal places): 4e 2x = 50 e 2x = 50/4 = 12.5 ln e 2x = 2x(ln e) = ln 12.5 x = (ln 12.5)/2 = 1.263

7. EXAMPLE 3 Solve (3-decimal places): ln (x-2) + ln (2x+3) = ln x 2 ln (x-2)(2x+3) = ln x 2 ln (2x 2 -x-6) = ln x 2 2x 2 -x-6 = x 2

8. EXAMPLE 3 continued 2x 2 -x-6 = x 2 x 2 -x-6 = 0 (x-3)(x+2) = 0 x-3 = 0 or x+2 = 0 x = 3 or x = -2

9. EXAMPLE 3 concluded ln (x-2) + ln (2x+3) = ln x 2 Domain x-2>0, yields x>2 2x+3 > 0, yields x > -3/2 x 2 > 0, yields x  0. So answer is x = 3.

10. ...the sky is the limit! STUDY AND WORK HARD...