Exponential & Logarithmic functions

1. Exponential & Logarithmic functions

2. Exponential Functionsy= ax ; 1 ≠ a > 0 ,that’s a is a positive fraction or a number greater than 1Case(1): a > 1 , Example: f(x) = 2xCase(2): 0<a <1 , Example: f(x) = (1/2)xg(x) = (1/2)x = 2-x = f(-x) is the reflection of f about the y-axis

3. f(x) = 2xdomain f = R , Range f = (0 , ∞ )f is everywhere continuous g(x) = (1/2)xdomain g = R , Range g = (0 , ∞ )g is everywhere continuous

4. Logarithmic Functionsh(x) = logax; 1 ≠ a > 0, (a is a positive fraction or a number greater than 1)Case(1): a > 1 Example: h(x) = log2x Let f(x) = 2xh(x) = log2x = f-1(x) = the reflection of f about the line y = x.

5. f(x) = 2xdomain f = R , Range f = (0 , ∞ )f is everywhere continuous y = 0 is a horizontal asymptote for f h(x) = log2xdomain h = (0 , ∞ ) , Range f = Rh is continuous on (0 , ∞ ) X = 0 is a vertical asymptote for h

6. Case(2): 0<a <1 Example: v(x) = log1/2x Let: h(x) = log2xv(x) = log1/2x = - log2x = the reflection of h about the x-axis

7. h(x) = log2xdomain h = (0 , ∞ ) , Range f = Rh is continuous on (0 , ∞ ) X = 0 is a vertical asymptote for h v(x) = log1/2xdomain v = (0 , ∞ ) , Range v = Rv is continuous on (0 , ∞ ) X = 0 is a vertical asymptote for v