Exponential & Logarithmic functions

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

f(x) = 2 x domain f = R, Range f = (0, ∞ ) f is everywhere continuous g(x) = (1/2) x = 2 -x The graph of g is the reflection of the graph of f about the y-axis domain g = R, Range g = (0, ∞ ) g is everywhere continuous

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

h(x) = log 2 x The graph of h is the reflection of the graph of f about the line y = x domain h = (0, ∞ ), Range f = R h is continuous on (0, ∞ ) X = 0 is a vertical asymptote for h f(x) = 2 x domain f = R, Range f = (0, ∞ ) f is everywhere continuous y = 0 is a horizontal asymptote for f

Case(2): 0<a <1 Example: v(x) = log 1/2 x Let: h(x) = log 2 x v(x) = log 1/2 x = - log 2 x = the reflection of h about the x-axis The graph of v is the reflection of the graph of h about the x-axis

h(x) = log 2 x domain h = (0, ∞ ), Range f = R h is continuous on (0, ∞ ) X = 0 is a vertical asymptote for h v(x) = log 1/2 x domain v = (0, ∞ ), Range v = R v is continuous on (0, ∞ ) X = 0 is a vertical asymptote for v