HCH3: Logarithmic & Exponential Functions

Question 1

Index laws

Answer 1

a^x x a^y = a^(x+y)
a^x ÷ a^y = a^(x-y)
(a^x)^y = a^(xy)
a^1 = a
a^0 = 1
a^-m = 1 ÷ a^m

Question 2

Logarithms

Answer 2

loga y = x

a^x = y

Question 3

Log laws

Answer 3

loga (mn) = loga m + loga n

loga (m/n) = loga m - loga n

loga (m^n) = n loga m

loga 1 = 0

loga a = 1

loga (1/x) = -loga x

Question 4

Change of base formula

Answer 4

loga b = In (b) ÷ In (a)

Question 5

Graph of y = a^x

Answer 5

• Exponential curve

- Always crosses y axis at (0, 1)

Question 6

Graph of y = log a x

Answer 6

• Inverse function of y= a^x
• Reflects y = a^x
• Passes through point (1, 0) and (a, 1)

Question 7

Derivative of e^x

Answer 7

If f(x) = e^x
   f ' (x) = e^x   x   d/dx (x)

Question 8

Integral of e^x

Answer 8

∫ e^(ax + b) dx

= 1÷ a e^(ax+ b) + C

Question 9

Natural logarithms

Answer 9

y = loge x

x = e^y

y = ln x

Question 10

Derivative of y = loge x

Answer 10

f ‘ (x) = f’(x) ÷ f (x)

It may be necessary to change the numerator so it is equal to the derivative of the denominator

If this is the case, you must then multiply by the reciprocal of that number to cancel it out

Question 11

Integral of 1/x

Answer 11

∫ 1/x dx = ln x + c

∫ f’ (x) ÷ f (x) dx = ln f (x) + C