Exponentials and Logarithms

Question 1

What is an exponential?

Answer 1

A function in the form y = a^x (or f(x) = a^x) where a > 0 and a is not equal to 1.

Question 2

What is the basic shape for an exponential graph when a > 1?

Answer 2

• All the graphs go through 1 at x = 0 since a^0 = 1 for any a.
• As y increases, x increases.
• The bigger a is, the quicker the graph increases (so the curve is curve is steeper).
• As x decreases, y decreases at a smaller and smaller rate so y will approach 0, but never actually get there.
• This means that as x tends to infinity, y tends to infinity and as x tends to negative infinity, y tends to 0.

Question 3

What is the basic shape for an exponential graph when 0 < a < 1?

Answer 3

• All the graphs go through 1 at x = 0 since a^0 = 1 for any a.
• As y decreases, x increases.
• The smaller a is, the faster the graph decreases (so the curve is steeper).
• As x increases, y decreases at a smaller and smaller rate so y will approach 0, but never actually get there.
• This means as x tends to infinity, y tends towards 0 and as x tends towards negative infinity, y tends towards infinity.

Question 4

What is e?

Answer 4

The value of ‘a’ for which the gradient of y = a^x is exactly the same as a^x. It is an irrational number around 2.7183. The exponential function y = e^x, is ‘the exponential function’.

Question 5

What is the gradient of the graph y = Ae^kx, where A and k are a constants?

Answer 5

kAe^kx

Question 6

What is a logarithm?

Answer 6

The power that a number needs to be raised to, to produce a given value.
e.g. log(a)b = c means the same as a^c = b

Question 7

How else can you write log(10)x?

Answer 7

log()x

Question 8

How else can you write log(e)x?

Answer 8

ln x (said ‘natural log of x’).

Question 9

For the function f(x) = a^x, what is the inverse?

Answer 9

log(a)x

Question 10

For the function f(x) = e^x, what is the inverse?

Answer 10

ln x

Question 11

What does the graph of an inverse function look like?

Answer 11

A reflection in the line y = x:

• y = ln x is the reflection of y = e^x in the line y = x.
• It cuts the x-axis at (1, 0) (so ln 1 = 0).
• as x tends to infinity, ln x tends towards infinity, but it happens very slowly.
• as x tends towards 0, ln x tends towards negative infinity.

Question 12

What are the three laws of logs?

Answer 12

• log(a)x + log(a)y = log(a)(xy)
• log(a)x - log(a)y = log(a)(x/y)
• log(a)x^k = klog(a)x

Question 13

What is the formula to change the base of a log?

Answer 13

log(a)x = (log(b)x)/(log(b)a)

Question 14

What is a^(kog(a)x)?

Answer 14

x

Question 15

What is log(a)a^x)?

Answer 15

x

Question 16

What is exponential growth?

Answer 16

When the rate of growth increases as the amount increases.

Question 17

What is exponential decay?

Answer 17

When the rate of decay decreases as the amount decreases.

Question 18

What is a logarithmic graph?

Answer 18

When you use logs to convert an equation of the form y = ax^n or y = kb^x to linear form so that you can plot this as a straight line graph, which is easier to work with.

Question 19

How do you convert y = ax^n to linear form?

Answer 19

logy = nlogx + loga

Question 20

For y = ax^n, how would you plot the linear graph?

Answer 20

Plot the values of logy on the y axis and logx on the x axis.

Question 21

How do you convert y = kb^x to linear form?

Answer 21

logy = xlogb + logk

Question 22

For y = ax^n, what is the gradient and y-intercept of the linear graph?

Answer 22

n is the gradient and loga is the y-intercept.

Question 23

For y = kb^x, what is the gradient and y-intercept of the linear graph?

Answer 23

logb is the gradient and log k is the y-intercept.

Question 24

For y = kb^x, how would you plot the linear graph?

Answer 24

Plot the values of logy on the y-axis and x on the x-axis.