How are exponential functions related to logarithmic functions? Model that with an example.
1 answer:
Exponential functions are related to logarithmic functions in that they are inverse functions. Exponential functions move quickly up towards a [y] infinity, bounded by a vertical asymptote (aka limit), whereas logarithmic functions start quick but then taper out towards an [x] infinity, bounded by a horizontal asymptote (aka limit).
If we use the natural logarithm (ln) as an example, the constant "e" is the base of ln, such that:
ln(x) = y, which is really stating that the base (assumed "e" even though not shown), that:
if we try to solve for y in this form it's nearly impossible, that's why we stick with ln(x) = y
but to find the inverse of the form:
switch the x and y, then solve for y:
So the exponential function is the inverse of the logarithmic one, f(x) = ln x
If we use the natural logarithm (ln) as an example, the constant "e" is the base of ln, such that:
ln(x) = y, which is really stating that the base (assumed "e" even though not shown), that:
if we try to solve for y in this form it's nearly impossible, that's why we stick with ln(x) = y
but to find the inverse of the form:
switch the x and y, then solve for y:
So the exponential function is the inverse of the logarithmic one, f(x) = ln x
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Answer:
Zeke will catch up with Niko at 16 yards from the start line, y = 16 yards.
Zeke will catch up Niko after 8 seconds, x = 8 seconds.
Step-by-step explanation:
i) distance to be run = 30 yards
ii) Niko has a head start of 12 yards.
iii) speed of Zeke = 2 yards / second
iv) speed of Niko = 1 yard every 2 seconds = 0.5 yard / second
v) x represents the number of seconds they have been racing.
vi) y represents the distance from the start line.
vii) the time at which Zeke catches up with Niko will be given by
Therefore 2y - 24 = 0.5y 1.5 y = 24
y = 24 / 1.5 = 16 yards
Therefore x = 16 /2 = 8 seconds