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.

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)

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.

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

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!

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).

why logarithms used in graphs ?

Interesting question as a fraction of people might get confused in it. Log is the inverse function of exponent which you can say another form of exponent...so like we use exponents in graph eg ---> y = x^2 same we can also write log_x (y) = 2 And both are the same but at some point things get easier while using log. A bit of a practice and experience and you will understand what I am saying. Have a good day !!

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.

If 2log(x+4)-log(x+12)=log10, I get x=-8,4. Do I have to eliminate -8 from the answer because the -8 makes (x+4) a negative number? Or I keep -8 because the coefficient attached to the log (x+4) turns (-4)^2=16?

You are correct that when x = -8, the expression (x + 4) becomes negative, which is not valid because the logarithm of a negative number is undefined. However, you should keep both answers (-8 and 4) and check them separately to see which one satisfies the original equation. When x = 4, 2log(x+4) - log(x+12) = 2log(8) - log(16) = 2(3) - 1 = 5, and log(10) = 1, so the left-hand side of the equation matches the right-hand side. Therefore, x = 4 is a valid solution. When x = -8, 2log(x+4) - log(x+12) = 2log(-4) - log(4) = undefined, so x = -8 is not a valid solution. Therefore, the only solution to the equation is x = 4.

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 !

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).

Main content

Algebra 2 (Eureka Math/EngageNY)

Course: Algebra 2 (Eureka Math/EngageNY) > Unit 3

Lesson 5: Topic B: Lessons 10-12: Logarithm properties

Intro to logarithm properties

Google Classroom

Learn about the properties of logarithms and how to use them to rewrite logarithmic expressions. For example, expand log₂(3a).


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
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
The power rule	log, 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 does all this mean again?]

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.

[Show me a numerical example of this property please.]

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.

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.

[Show me a numerical example of this property please.]

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.

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.

[Show me a numerical example please.]

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}

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?

(Choice A)
3, log, start base, b, end base, left parenthesis, 2, x, right parenthesis, minus, log, start base, b, end base, left parenthesis, 5, right parenthesis
(Choice B)
3, log, start base, b, end base, left parenthesis, 2, x, right parenthesis, plus, log, start base, b, end base, left parenthesis, 5, right parenthesis
(Choice C)
log, start base, b, end base, left parenthesis, 2, right parenthesis, plus, 3, log, start base, b, end base, left parenthesis, x, right parenthesis, minus, log, start base, b, end base, left parenthesis, 5, right parenthesis
(Choice D)
log, start base, b, end base, left parenthesis, 2, right parenthesis, minus, 3, log, start base, b, end base, left parenthesis, x, right parenthesis, minus, log, start base, b, end base, left parenthesis, 5, 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?

(Choice A)
log, start base, 2, end base, left parenthesis, start fraction, x, cubed, divided by, 25, end fraction, right parenthesis
(Choice B)
start fraction, log, start base, 2, end base, left parenthesis, x, cubed, right parenthesis, divided by, log, start base, 2, end base, left parenthesis, 25, right parenthesis, end fraction
(Choice C)
minus, 6, log, start base, 2, end base, left parenthesis, start fraction, x, divided by, 5, end fraction, right parenthesis
(Choice D)
minus, 6, log, start base, 2, end base, left parenthesis, x, minus, 5, right parenthesis

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Blue
Posted 7 years ago. Direct link to Blue's post “Why isn't there a base in...”
Why isn't there a base in question 4?
Button navigates to signup pageComment on Blue's post “Why isn't there a base in...”
(26 votes)
Answer
- Vu
  Posted 7 years ago. Direct link to Vu's post “Since both of them have n...”
  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.
  Button navigates to signup page
  (100 votes)
Rakhi Kumari
Posted 6 years ago. Direct link to Rakhi Kumari's post “in the 1st challenging pr...”
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
Button navigates to signup pageButton navigates to signup page
(9 votes)
Answer
- Judith Gibson
  Posted 6 years ago. Direct link to Judith Gibson's post “Your mistake is in dealin...”
  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)
  Comment on Judith Gibson's post “Your mistake is in dealin...”
  (45 votes)
Esha
Posted 7 years ago. Direct link to Esha's post “why the base cant be 1 of...”
why the base cant be 1 of a logarithm?
Button navigates to signup pageComment on Esha's post “why the base cant be 1 of...”
(6 votes)
Answer
- Vu
  Posted 7 years ago. Direct link to Vu's post “Let take an example of lo...”
  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.
  Button navigates to signup page
  (45 votes)
Samia
Posted 6 years ago. Direct link to Samia's post “Can the answer to logarit...”
Can the answer to logarithm be negative?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Howard Bradley
  Posted 6 years ago. Direct link to Howard Bradley's post “Any number less than 1 bu...”
  Any number less than 1 but greater than 0 will have a negative logarithm*
  eg ln(0.1) = -2.303
  Comment on Howard Bradley's post “Any number less than 1 bu...”
  (17 votes)
April H.
Posted 3 years ago. Direct link to April H.'s post “Is there a distributive p...”
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!
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- kubleeka
  Posted 3 years ago. Direct link to kubleeka's post “Properties like this are ...”
  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).
  Comment on kubleeka's post “Properties like this are ...”
  (8 votes)
Randy Smart
Posted 5 years ago. Direct link to Randy Smart's post “It looks like the answer ...”
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 ?
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Alex Evangelopoulos
  Posted 3 years ago. Direct link to Alex Evangelopoulos's post “The three exponent only r...”
  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).
  Button navigates to signup page
  (9 votes)
Mahmoud Taha Tanab
Posted 2 months ago. Direct link to Mahmoud Taha Tanab's post “why logarithms used in gr...”
why logarithms used in graphs ?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Arnav Kumar
  Posted 2 months ago. Direct link to Arnav Kumar's post “Interesting question as a...”
  Interesting question as a fraction of people might get confused in it.
  
  Log is the inverse function of exponent which you can say another form of exponent...so like we use exponents in graph eg ---> y = x^2 same we can also write log_x (y) = 2
  And both are the same but at some point things get easier while using log.
  
  A bit of a practice and experience and you will understand what I am saying.
  
  Have a good day !!
  Button navigates to signup page
  (3 votes)
crishawn-waden
Posted 3 years ago. Direct link to crishawn-waden's post “where is the base”
where is the base
Button navigates to signup pageComment on crishawn-waden's post “where is the base”
(2 votes)
Answer
- A/V
  Posted 3 years ago. Direct link to A/V's post “logₐ(b) = a is your base ...”
  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 !
  Button navigates to signup page
  (5 votes)
Rachel M
Posted 3 years ago. Direct link to Rachel M's post “When condensing and the o...”
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 :)
Button navigates to signup pageComment on Rachel M's post “When condensing and the o...”
(3 votes)
Answer
- Kim Seidel
  Posted 3 years ago. Direct link to Kim Seidel's post “"Log base e" and "ln" are...”
  "Log base e" and "ln" are the same. You can use either one, though using "ln" is less to write.
  Button navigates to signup page
  (2 votes)
krldec1224
Posted 4 months ago. Direct link to krldec1224's post “If 2log(x+4)-log(x+12)=lo...”
If 2log(x+4)-log(x+12)=log10, I get x=-8,4.
Do I have to eliminate -8 from the answer because the -8 makes (x+4) a negative number? Or I keep -8 because the coefficient attached to the log (x+4) turns (-4)^2=16?
Button navigates to signup pageButton navigates to signup page
(3 votes)
Answer
- Hazel
  Posted 3 months ago. Direct link to Hazel's post “You are correct that when...”
  You are correct that when x = -8, the expression (x + 4) becomes negative, which is not valid because the logarithm of a negative number is undefined. However, you should keep both answers (-8 and 4) and check them separately to see which one satisfies the original equation.
  
  When x = 4, 2log(x+4) - log(x+12) = 2log(8) - log(16) = 2(3) - 1 = 5, and log(10) = 1, so the left-hand side of the equation matches the right-hand side. Therefore, x = 4 is a valid solution.
  
  When x = -8, 2log(x+4) - log(x+12) = 2log(-4) - log(4) = undefined, so x = -8 is not a valid solution.
  
  Therefore, the only solution to the equation is x = 4.
  Button navigates to signup page
  (2 votes)

Algebra 2 (Eureka Math/EngageNY)

Course: Algebra 2 (Eureka Math/EngageNY) > Unit 3

What you should be familiar with before taking this lesson

What you will learn in this lesson

Example: Expanding logarithms using the product rule

Example: Condensing logarithms using the product rule

An important note

Check your understanding

Example: Expanding logarithms using the quotient rule

Example: Condensing logarithms using the quotient rule

An important note

Check your understanding

The power rule: log⁡b(Mp)=plog⁡b(M)\log_b(M^p)=p\log_b(M)logb​(Mp)=plogb​(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

Example: Expanding logarithms using the power rule

Example: Condensing logarithms using the power rule

Check your understanding

Challenge problems

Want to join the conversation?

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