Chapter 1. Logs and exponentials
6.1 - Introduction to Exponents
The notation xn stands for the quantity obtained by multiplying x by itself n times. For example, x2 = x . x and x3 = x . x . x
The quantity n is called the power, or the exponent, of x (called the base).
Throughout the book you will come across different examples of equations that use exponents, listed below are some rules that will help you simplify terms that
have exponents.
Rule 1. To multiply two different powers of x, add the exponents.
(xm)(xn) = x(m+n)
Example: x2 . x3 = x(2+3) = x5
We can see this result must be true by expanding the different exponents out.
x2 = x.x and x3 = x.x.x so x2.x3 = (x.x)(x.x.x) - there are five ‘x’ here multiplied together – which can be written as x5
Rule 2: Any number to the power 0 is equal to 1.
x0 = 1
This also means that if adding exponents (e.g. applying rule 1) produces a zero, then the answer is equal to 1.
Example:
x-n.xn = x0 = 1
Note: remember that 1/xn = x-n
Rule 3: When two powers are divided, the exponents should be subtracted.
xn/xm = xnx-m = x(n-m)
Example:
x6/x2 = x(6-2) = x4
Rule 4: When a power is raised to another power, the exponents are multiplied
(xn)m = xnm
Example:
(x3)2 = x6
Again expanding the exponents out shows that this must be true.
x3 = x.x.x and x2 = x.x so (x3)2 = (x.x.x)(x.x.x) - there are five ‘x’ here multiplied together – which can be written as x6
Rule 5: When exponents are written as fractions, they represent the roots of the base.
For example,
x1/2 = √x and x1/3 = 3√x (This is only true if x>1)
It follows then that x1/2 .x1/2 = x1 = x
P’Cast: Simplifying a Quantity That Has Exponents
Try it yourself 1 - Exponents
In this section you can practice the rules for exponents by simplifying these expressions:
(Try to solve each one then click to reveal the solution)
1. x2x5
answer = x7
2. x4x
answer = x5
3. x0
answer = 1
4. 1/x2
answer = x-2
5. x5/x2
answer = x3
6. (x3x2)2
answer = x10
7. x-2x4
answer = x2
8. x-3/x2
answer = x-5
9. x0x-1x2
answer = x
10. (x1/3)3
answer = x
6.2. Introduction to Logarithms
Logarithms are closely related to the exponents we discussed in the previous section. Here we will see that there are also a few rules that can be used to simplify equations that include logarithms.
What are Logarithms?
Any positive number, y can be expressed as some power of any other positive number, x, except the number 1.
Based on this idea, we can write down a general statement about how y is related to x,
y = ax,
The number x is said to be the “logarithm of y to the base a”, and another way to write the relation is
x = loga y
Thus, logarithms are exponents, and the rules for working with logarithms are similar to those for exponents. Listed next are some rules that will help you simplify equations with logarithms – notice the similarities between these rules and those for exponents.
Rule 1: In the section on exponents earlier we learned that
If y1 = an and y2 = am,
then
y1 y2 = anam = an+m
Now if I take the log of both sides of this equation, we get
loga (y1 y2) = loga an+m
Since we know that x = loga y (our initial definition)
loga an+m must equal n+m
n + m = loga an + logaam = logay1 + logay2
So to summarize:
loga(y1y2) = logay1 + logay2
Another example:
log10 (x.z) = log10x +log10 z
There is a similar rule for fractions,
loga(x/y) = logax – loga y
Rule 2: Dealing with exponents inside a logarithm
If there is an exponent inside a logarithm, it can be factored out like this,
logxy = ylogx
Or
logax-2 = -2logax
Rule 3:The logarithm of 1 is equal to zero
This rule reflects the fact that any number raised to the power 0 is equal to 1.
Because a1 = a and a0 = 1,
loga a = 1 and loga 1 = 0
6.3. Frequently used Logarithms
There are two bases in common use: logarithms to base 10 are called common logarithms, and logarithms to base e (where e = 2.718 . . . ) are called natural logarithms.
In this text, the symbol “ln” is used for natural logarithms and the symbol “log”, without a subscript, is used for common logarithms.
Thus,
loge x = ln x
and
log10 x = log x
If y = ln x then x = ey
If y = log x, then x = 10y
FAQ: What if I want to change the base of a logarithm in an equation?
Logarithms can be changed from one base to another.
Suppose that
z = log x
Then we can write that
10z = 10log x = x
Taking the natural logarithm of both sides of Equation, we obtain
z ln 10 = ln x
Since we know that z = log x, substituting log x for z gives
ln x = (ln 10)log x
P’Cast: Converting between Common Logarithms and Natural Logarithms
Try it yourself 2 – logarithms
Now practice using the rules for logarithms by simplifying these expressions:
(Try to solve each one then click to reveal the solution)
1. log abn
(answer = loga + nlogb)
2. log x2 +log x2
(answer = 4logx)
3. ln 1 + ln e
(answer = 1)
4. ln y3/ln x3
(answer = ln x/ln y)
5. log 10x2
(answer = 1 + 2logx)