Log Rules

Log rules refer to the rules of logarithms. These rules are derived from the rules of exponents as a logarithm is just the other way of writing an exponent. The logarithm rules are used:

• to compress a group of logarithms into a single logarithm
• to expand a logarithm into a group of logarithms

Let us learn more about log rules and we will solve some example problems using the logarithm rules.

1.	What are Log Rules?
2.	Natural Log Rules
3.	Product Rule of Logarithms
4.	Quotient Rule of Logarithms
5.	Logarithm Power Rule
6.	Change of Base Rule of Logs
7.	FAQs on Log Rules

Log rules are rules that are used to operate logarithms. Since logarithm is just the other way of writing an exponent, we use the rules of exponents to derive the logarithm rules. There are mainly 4 important log rules which are stated as follows:

• product rule: log_b mn = log_b m + log_b n
• quotient rule: log_b m/n = log_b m - log_b n
• power rule: log_b mⁿ = n log_b m
• change of base rule: log_a b = (log_c b) / (log_c a)

The following log rules are derived from the formula of logarithmic form to exponential form and vice versa (b^x = m ⇔ log_b m = x).

• b⁰ = 1 ⇒ log_b 1 = 0
• b¹ = b ⇒ log_b b = 0

Logarithm Rules

Along with these rules, we have several other rules of logarithms. All logarithm rules are mentioned below:

Going forward, we will see how each of these rules is derived using the exponent rules.

A natural log is a logarithm with the base "e". It is denoted by "ln". i.e., log_e = ln. i.e., we do NOT write a base for the natural logarithm. When "ln" is seen automatically it is understood that its base is "e". The rules of logs are the same for all logarithms including the natural logarithm. Hence, the important natural log rules (rules of ln) are as follows:

• ln (mn) = ln m + ln n
• ln (m/n) = ln m - ln n
• ln mⁿ = n ln m
• ln a = (log a) / (log e)
• ln e = 1
• ln 1 = 0

The number raised to log rule (mentioned in the above table) is b^log_bx = x. The equivalent rule of ln is, e^{ln x} = x.

By the product rule of logarithms, the log of a product of two terms is equal to the sum of logs of individual terms. i.e., the rule says log_b mn = log_b m + log_b n. Let us derive this rule.

Derivation:

Assume that log_b m = x and log_b n = y. Changing each of these into exponential forms, we get

log_b m = x ⇒ m = b^x ... (1)

log_b n = y ⇒ n = b^y ... (2)

Multiply the equations (1) and (2),

mn = b^x · b^y

By using the multiplication rule of exponents,

mn = b^{x + y}

Changing this back to logarithmic form,

log_b mn = x + y

Substituting the values of x and y back here,

log_b mn = log_b m + log_b n

Hence, the product rule of logarithm is derived. We can apply this rule in the following ways:

• log (3a) = log 3 + log a
• log 10 = log (5×2) = log 5 + log 2
• log₃ (ab) = log₃ a + log₃ b

Note: The product rule of logs doesn't mean anything about log (m + n). It talks only about log (mn). log (m + n) cannot be further split into two logarithms indeed.

By the quotient rule of logarithms, the log of a quotient of two terms is equal to the difference of logs of individual terms. i.e., the rule says log_b mn = log_b m + log_b n. Let us derive this rule.

Derivation:

Assume that log_b m = x and log_b n = y. Let us convert these into exponential forms.

• log_b m = x ⇒ m = b^x ... (1)
• log_b n = y ⇒ n = b^y ... (2)

Dividing the equations (1) and (2),

m/n = b^x / b^y

By using the quotient rule of exponents,

m/n = b^{x - y}

Changing this back to logarithmic form,

log_b m/n = x - y

Substituting the values of x and y back here,

log_b m/n = log_b m - log_b n

Hence, the quotient rule of logarithm is derived. We can apply this rule in the following ways:

• log (y/3) = log y - log 3
• log 25 = log (125/3) = log 125 - log 3
• log₇ (a/b) = log₇ a - log₇ b

Note: The quotient rule does NOT mean anything about log (m - n).

The power rule of logs says that if the argument of a logarithm has an exponent, then the exponent can be brought to in front of the logarithm. i.e., log_b mⁿ = n log_b m. Let us derive this rule.

Derivation:

Assume that log_b m = x. Changing this into exponential form,

b^x = m

Raising both sides by n,

(b^x) ⁿ = mⁿ

By the power rule of exponents,

b^nx = mⁿ

Changing this into logarithmic form,

log_b mⁿ = nx

Substituting x = log_b m back here,

log_b mⁿ = n log_b m

Hence, the power rule of logarithms is derived. Here are a few applications of this rule.

• log 3^z = z log 3
• log y² = 2 log y
• log₃ y^x = x log₃ y

The change of base rule of logs, as its name suggests, is used to change the base of a logarithm. It says, log_a b = (log_c b) / (log_c a). We can clearly see that the base on the left side of the rule is 'a' whereas the bases of logarithms on the right side are equal to 'c'. Let us derive this rule now.

Derivation:

Assume that log_a b = x, log_c b = y, and log_c a = z. Let us convert these equations into logarithmic form.

• log_a b = x ⇒ b = a^x ... (1)
• log_c b = y ⇒ b = c^y ... (2)
• log_c a = z ⇒ a = c^z ... (3)

From (1) and (2),

a^x = c^y

From (3), a = c^z. Substituting this in the above equation,

(c^z)^x = c^y

c^zx = c^y

zx = y (or) x = y/z

Substituting the values of x, y, and z here:

log_a b = (log_c b) / (log_c a)

Hence, the change of base rule of logs is derived. By multiplying it on both sides by log_c a, we get another form of change of base rule.

log_a b · log_c a = log_c b

Here are some applications of both forms of this rule:

• log₅ 3 = (log 3)/(log 5)
• log_x 4 · log₃ x = log₃ 4

The change of base rule is especially used when a logarithm (whose base is other than 10 and e) is to be calculated using a calculator. Notice that we have only "log" and "ln" buttons on the calculator. So to calculate a log with any other base, say log₅ 3, there is no button with base 5. Then by the change of base rule, log₅ 3 = (log 3)/(log 5) and this can be easily calculated using the "log" button of the calculator.

Important Notes on Logarithm Rules

• The logarithm rules are the same for both natural and common logarithms (log, log_a, and ln). The base of the log just carries to every log while applying the rules.
• log_a 1 = 0 for any base 'a'.
• The most commonly logarithm rules are:
  log_b mn = log_b m + log_b n
  log_b m/n = log_b m - log_b n
  log_b mⁿ = n log_b m

☛ Related Topics:

• Log Table
• Antilog Table
• Log Calculator
• Natural Log Calculator

 

What are the 7 Log Rules?

The log rules are used to expand or compress logarithms. The 7 commonly used log rules are mentioned in the table below.

Rule Name	Log Rule
Log 1	logb 1 = 0
Log of a Number With Same Base	logb b = 1
Product Rule	logb mn = logb m + logb n
Quotient Rule	logb m/n = logb m - logb n
Power Rule of Logarithm	logb mn = n logb m
Change of Base Rule	logb a = (log a) / ( log b)
Number Raised to Log	blogbx = x

When to Use Logarithm Rules?

When we need to expand a logarithm into multiple logarithms or compress multiple logarithms into a single logarithm, we use the logarithm rules. These rules are derived from the rules of exponents.

What is the Product Rule of Logarithms?

The product rule of logarithms is log_b mn = log_b m + log_b n. Using this rule, the logarithm of a product can be converted into a sum of logarithms. For example: log₂ (xyz) = log₂ x + log₂ y + log₂ z.

How to Convert a Negative Log into a Positive Log?

A negative log can be converted into a positive log using one of the following log rules:

• The negative log of an argument is the logarithm of the reciprocal of the argument. i.e., -log_b a = log_b a^-1 = log_b (1/a).
• The negative log with a base is the logarithm whose base is the reciprocal of the given base. i.e., -log_b a = log_(1/b) a.

What are the Rules of Ln?

The rules of "ln" deal with the natural logarithms. The natural logarithms are the logarithms with the base 'e'. The following are the rules of ln.

• ln 1 = 0
• ln e = 1
• ln m + ln n = ln (mn)
• ln m - ln n = ln (m/n)
• ln a^m = m ln a
• e^{ln x} = x

What are Log Derivative Rules?

Here are the derivatives of different types of logarithms:

• The derivative of ln x is, d/dx (ln x) = 1/x.
• The derivative of log_a x, d/dx (log x) = 1/(x ln a)

What are 4 Important Logarithm Rules?

We have many logarithm rules. Among them, the 4 important rules of common logs are as follows:

• log m + log n = log (mn)
• log m - log n = log (m/n)
• log a^m = m log a
• 10^{log x} = x

What is the Difference Between Log Rules and Natural Log Rules?

In fact, there is no difference between the rules of common logarithms and the rules of natural logarithms. This is because a natural log is also a logarithm (just with base 'e').