Exponentials & Logarithms (P1.14)

Question 1

sketch graph in the form y = a^x & transformations

Answer 1

exponential function

when a > 1, f(x) = a^x is an increasing function
as x increases, a^x grows without limit & as x decreases, it tends towards 0

when 0 < a < 1
a^x is a decreasing function
as x decreases, a^x grows without limit & as x increases, it tends towards

as a increases, f(x) goes closer to 0 on -ve x-axis & is steeper on +ve x-axis

Question 2

sketch graph in the form y = e^x & transformations

Answer 2

crosses y-axis at 1

Question 3

f(x) = e^kx
f’(x) =

Answer 3

ke^kx
for e^x, gradient function is the same as the original function

Question 4

describe exponential modelling

Answer 4

rate of increase is proportional to the size of population at any given moment, using e^x
use e^-x to model situations of decay, where the rate of decrease is proportional to the number of things (atoms) remaining

Question 5

changing form of logarithms

Answer 5

a^x = n
loga(n) = x

Question 6

what are the laws of logarithms

Answer 6

loga(x) + loga(y) = loga(xy)
loga(x) - loga(y) = loga(x/y)
loga(x^k) = kloga(x)

loga(1/x) = -loga(x)
loga(a) = 1
loga(1) = 0

Question 7

describe how to solve equations using logarithms

Answer 7

change form / take logs of both sides
f(x) = g(x)
logaf(x) = logag(x)

Question 8

describe the graph of y = lnx

Answer 8

y = lnx is a reflection of the graph y = e^x in the line y = x
y = lnx passes through (1,0) & has asymptote at y-axis –> lnx is only defined for +ve values of x
as x increases, lnx grows without limit, but relatively slowly
e^lnx = ln(e^x) = x

Question 9

how can logarithms be used to to manage non-linear trends in data?

Answer 9

y = ax^n
y = ab^x

Question 10

convert y = ax^n into log form

Answer 10

y = ax^n
logy = log(ax^n)
logy = loga + log(x^n)
logy = loga + nlogx
compare to y = mx + c

the graph of logy against logx will be a straight line with gradient n & vertical intercept loga

Question 11

convert y = ab^x into log form

Answer 11

y = ab^x
logy = log(ab^x)
logy = loga + log(b^x)
logy = loga + xlogb
compare to y = mx + c

the graph of logy against x will be a straight line with gradient logb & vertical intercept loga