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

(4,-.5), (16,0) are points on the graph of a log equation f(t) = a+logb(t)

a=-1, b=16 base

0 = a+logb(16)

-.5 = a +logb(4)

two equations, two unknowns

-a = logb(16),

-.5-a= logb(4)

subtract

.5= logb(16)- logb(4)

logb(16/4)=1/2

sqrb=4

b=16

a=-log16(16)=-1

f(t)= -1+ln16(t)

Answer 2

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.