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# Intro to logarithm properties

Learn about the properties of logarithms and how to use them to rewrite logarithmic expressions. For example, expand log₂(3a).
The product rulelog, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis
The quotient rulelog, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis
The power rulelog, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, dot, log, start base, b, end base, left parenthesis, M, right parenthesis
(These properties apply for any values of M, N, and b for which each logarithm is defined, which is M, N, is greater than, 0 and 0, is less than, b, does not equal, 1.)

#### What you should be familiar with before taking this lesson

You should know what logarithms are. If you don't, please check out our intro to logarithms.

#### What you will learn in this lesson

Logarithms, like exponents, have many helpful properties that can be used to simplify logarithmic expressions and solve logarithmic equations. This article explores three of those properties.
Let's take a look at each property individually.

## The product rule: $\log_b(MN)=\log_b(M)+\log_b(N)$log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis

This property says that the logarithm of a product is the sum of the logs of its factors.
We can use the product rule to rewrite logarithmic expressions.

### Example: Expanding logarithms using the product rule

For our purposes, expanding a logarithm means writing it as the sum of two logarithms or more.
Let's expand log, start base, 6, end base, left parenthesis, 5, y, right parenthesis.
Notice that the two factors of the argument of the logarithm are start color #11accd, 5, end color #11accd and start color #1fab54, y, end color #1fab54. We can directly apply the product rule to expand the log.
\begin{aligned} \log_6(\blueD5\greenD y)&=\log_6(\blueD5\cdot \greenD y) \\\\ &=\log_6(\blueD5)+\log_6(\greenD y)&&{\gray{\text{Product rule}}} \end{aligned}

### Example: Condensing logarithms using the product rule

For our purposes, compressing a sum of two or more logarithms means writing it as a single logarithm.
Let's condense log, start base, 3, end base, left parenthesis, 10, right parenthesis, plus, log, start base, 3, end base, left parenthesis, x, right parenthesis.
Since the two logarithms have the same base (base-3), we can apply the product rule in the reverse direction:
\begin{aligned} \log_3(\blueD{10})+\log_3(\greenD x)&=\log_3(\blueD{10}\cdot \greenD x)&&{\gray{\text{Product rule}}} \\\\ &=\log_3({10} x) \end{aligned}

### An important note

When we compress logarithmic expressions using the product rule, the bases of all the logarithms in the expression must be the same.
For example, we cannot use the product rule to simplify something like log, start base, 2, end base, left parenthesis, 8, right parenthesis, plus, log, start base, 3, end base, left parenthesis, y, right parenthesis.

1) Expand log, start base, 2, end base, left parenthesis, 3, a, right parenthesis.

2) Condense log, start base, 5, end base, left parenthesis, 2, y, right parenthesis, plus, log, start base, 5, end base, left parenthesis, 8, right parenthesis.

## The quotient rule: $\log_b\left(\dfrac{M}{N}\right)=\log_b(M)-\log_b(N)$log, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis

This property says that the log of a quotient is the difference of the logs of the dividend and the divisor.
Now let's use the quotient rule to rewrite logarithmic expressions.

### Example: Expanding logarithms using the quotient rule

Let's expand log, start base, 7, end base, left parenthesis, start fraction, a, divided by, 2, end fraction, right parenthesis, writing it as the difference of two logarithms by directly applying the quotient rule.
\begin{aligned} \log_7\left(\dfrac{\purpleC a}{\goldD 2}\right)&=\log_7(\purpleC a)-\log_7(\goldD 2) &{\gray{\text{Quotient rule}}} \end{aligned}

### Example: Condensing logarithms using the quotient rule

Let's condense log, start base, 4, end base, left parenthesis, x, cubed, right parenthesis, minus, log, start base, 4, end base, left parenthesis, y, right parenthesis.
Since the two logarithms have the same base (base-4), we can apply the quotient rule in the reverse direction:
\begin{aligned} \log_4(\purpleC{x^3})-\log_4(\goldD{y})&=\log_4\left(\dfrac{\purpleC{x^3}}{\goldD{y}}\right)&&{\gray{\text{Quotient rule}}} \end{aligned}

### An important note

When we compress logarithmic expressions using the quotient rule, the bases of all logarithms in the expression must be the same.
For example, we cannot use the quotient rule to simplify something like log, start base, 2, end base, left parenthesis, 8, right parenthesis, minus, log, start base, 3, end base, left parenthesis, y, right parenthesis.

3) Expand log, start base, b, end base, left parenthesis, start fraction, 4, divided by, c, end fraction, right parenthesis.

4) Condense log, left parenthesis, 3, z, right parenthesis, minus, log, left parenthesis, 8, right parenthesis.

## The power rule: $\log_b(M^p)=p\log_b(M)$log, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis

This property says that the log of a power is the exponent times the logarithm of the base of the power.
Now let's use the power rule to rewrite log expressions.

### Example: Expanding logarithms using the power rule

For our purposes in this section, expanding a single logarithm means writing it as a multiple of another logarithm.
Let's use the power rule to expand log, start base, 2, end base, left parenthesis, x, cubed, right parenthesis.
\begin{aligned} \log_2\left(x^\maroonC3\right)&=\maroonC3\cdot \log_2(x)&&{\gray{\text{Power rule}}} \\\\ &=3\log_2(x) \end{aligned}

### Example: Condensing logarithms using the power rule

For our purposes in this section, condensing a multiple of a logarithm means writing it as another single logarithm.
Let's use the power rule to condense 4, log, start base, 5, end base, left parenthesis, 2, right parenthesis,
When we condense a logarithmic expression using the power rule, we make any multipliers into powers.
\begin{aligned} \maroonC4\log_5(2)&=\log_5\left(2^\maroonC 4\right)&&{\gray{\text{Power rule}}} \\\\ &=\log_5(16) \end{aligned}

5) Expand log, start base, 7, end base, left parenthesis, x, start superscript, 5, end superscript, right parenthesis.

6) Condense 6, natural log, left parenthesis, y, right parenthesis.

## Challenge problems

To solve these next problems, you will have to apply several properties in each case. Give it a try!
7) Which of the following is equivalent to log, start base, b, end base, left parenthesis, start fraction, 2, x, cubed, divided by, 5, end fraction, right parenthesis?

8) Which of the following is equivalent to 3, log, start base, 2, end base, left parenthesis, x, right parenthesis, minus, 2, log, start base, 2, end base, left parenthesis, 5, right parenthesis?

## Want to join the conversation?

• Why isn't there a base in question 4?
• Since both of them have no base you can presume they have the same base (no base) so you can apply the property of log. Your answer would be without base as well.

Later on you'll learn that no base log is understood as base of 10.
• why the base cant be 1 of a logarithm?
• Let take an example of log₁(25)=x
This would mean 1^x=25
But 1 to any powers will always be 1.
So the base can't be 1 because it would make the log expression false, unless log₁(1)=x but then x would be any and all real numbers. So the convention is to rule out log base 1.
• in the 1st challenging problem the ans given is (c) instead of a.if simplified
we get log base(b) (2x^3/5)= 3log base(b) 2+3log base(b) x - log base(b) 5. it shud be either a
• Your mistake is in dealing with log_b (2x^3).
It is only the x that is cubed, not the 2.
So we first need to separate the factors (2 and x^3), getting
log_b (2) + log_b (x^3).
Only then can we deal with the cube, getting
log_b (2) + 3 log_b (x)
• Can the answer to logarithm be negative?
• Any number less than 1 but greater than 0 will have a negative logarithm*
eg ln(0.1) = -2.303
• It looks like the answer to challenge question is wrong if "C" is answer.
Should be 3log_B(2) + 3log_B(X) - log_B(5).

Answer "C" leaves out the multiplier "3" in the first term of the solution.
Right ?
• The three exponent only refers to the x, ao you can't expand using the power rule until you first use the product rule to get rid of the 2: log_b(2) + log_b(x^3) - log_b(5) and then expand the x^3 to get log_b(2) + 3log_b(x) - log_b(5).
• Is there a distributive property for logarithms? For example, does ln(a+b)=ln(a)+ln(b)? I don't think I would use the product rule here. Is this covered in another lesson? Thanks!
(1 vote)
• Properties like this are covered later in the logarithms playlist. The property you suggest doesn't hold. ln(1+2)=ln(3), but ln(1)+ln(2)=0+ln(2)=ln(2), and ln(3)≠ln(2).

The relevant property here is that ln(ab)=ln(a)+ln(b). If you start from the property e^xe^y=e^(x+y), you can take the natural log of both sides to get

ln(e^xe^y)=x+y

Now let e^x=a (or x=ln(a)) and e^y=b (or y=ln(b)) and substitute in to get ln(ab)=ln(a)+ln(b).
• When condensing and the original problem begins with ln, do you need to switch it to log base e or can you leave it as ln?

ex: 2 ln 5 - 3 ln 2

final answer: log base e (25/8) or ln (25/8)? thank you :)
• "Log base e" and "ln" are the same. You can use either one, though using "ln" is less to write.
• What happens when there is an exponent outside of parentheses that have more than one term in them? EX: log(a+x^2)^4

My teacher told me that it is distributed throughout, but do I multiply or add the exponents of problems like these? Thank you!
• where is the base
(1 vote)
• logₐ(b) = a is your base

if it's simply just log(b), always assume 10 is your base
if it's ln(b), always assume that "e" (an irrational constant like pi) is the base.

hopefully that helps !