Find the formula for the logarithmic function.

Question

Find the formula for the logarithmic function of the form:

f(t) = a + logb(t)

Whose graph contains the points (4, -0.5) and (16, 0)

Answer 1

f(t) = a + log_bt

f(4) = -0.5 So, a + log_b4 = -0.5

f(16) = 0 So, a + log_b16 = 0

Subtract the equations to get log_b4 - log_b16 = -0.5.

log_b(4/16) = -0.5 So, 1/4 = b^-0.5. 1/4 = 1 / √b Therefore, b = 16.

a + log₁₆16 = 0 So, a + 1 = 0. Thus, a = -1.