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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, 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.
log6(5y)=log6(5y)=log6(5)+log6(y)Product rule\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:
log3(10)+log3(x)=log3(10x)Product rule=log3(10x)\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.

Check your understanding

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, 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.
log7(a2)=log7(a)log7(2)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:
log4(x3)log4(y)=log4(x3y)Quotient rule\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.

Check your understanding

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, 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.
log2(x3)=3log2(x)Power rule=3log2(x)\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.
4log5(2)=log5(24)Power rule=log5(16)\begin{aligned} \maroonC4\log_5(2)&=\log_5\left(2^\maroonC 4\right)&&{\gray{\text{Power rule}}} \\\\ &=\log_5(16) \end{aligned}

Check your understanding

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?
Choose 1 answer:
Choose 1 answer:

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?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • aqualine ultimate style avatar for user Blue
    Why isn't there a base in question 4?
    (23 votes)
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    • duskpin sapling style avatar for user Vu
      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.
      (91 votes)
  • leafers ultimate style avatar for user Esha
    why the base cant be 1 of a logarithm?
    (6 votes)
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    • duskpin sapling style avatar for user Vu
      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.
      (43 votes)
  • marcimus pink style avatar for user Rakhi Kumari
    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
    (6 votes)
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  • duskpin ultimate style avatar for user Samia
    Can the answer to logarithm be negative?
    (3 votes)
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  • starky seedling style avatar for user Randy Smart
    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 ?
    (0 votes)
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  • blobby green style avatar for user April H.
    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)
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    • leaf green style avatar for user kubleeka
      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).
      (6 votes)
  • duskpin ultimate style avatar for user Rachel M
    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 :)
    (2 votes)
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  • aqualine ultimate style avatar for user 22brhunt
    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!
    (2 votes)
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  • blobby green style avatar for user crishawn-waden
    where is the base
    (1 vote)
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    • leaf orange style avatar for user A/V
      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 !
      (4 votes)
  • stelly blue style avatar for user Surabhi
    Is there any property we can use when multiplying logs with same bases?
    eg. in, log_2 of 8 * log_2 of 4?
    (2 votes)
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