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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 
What are 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} m^{n} = 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).
 b^{0 }= 1 ⇒ log_{b} 1 = 0
 b^{1} = 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.
Natural Log 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 m^{n} = 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^{logbx} = x. The equivalent rule of ln is, e^{ln x} = x.
Product Rule of Logarithms
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
 log_{3} (ab) = log_{3} a + log_{3} 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.
Quotient Rule of Logarithms
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
 log_{7} (a/b) = log_{7} a  log_{7} b
Note: The quotient rule does NOT mean anything about log (m  n).
Logarithm Power Rule
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} m^{n} = 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} = m^{n}
By the power rule of exponents,
b^{nx }= m^{n}
Changing this into logarithmic form,
log_{b} m^{n} = nx
Substituting x = log_{b} m back here,
log_{b} m^{n} = 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 y^{2} = 2 log y
 log_{3} y^{x} = x log_{3} y
Change of Base Rule of Logs
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:
 log_{5} 3 = (log 3)/(log 5)
 log_{x} 4 · log_{3} x = log_{3} 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 log_{5} 3, there is no button with base 5. Then by the change of base rule, log_{5} 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} m^{n} = n log_{b} m
☛ Related Topics:
Logarithm Rules Examples

Example 1: Compress each of the following into a single logarithm. a) log_{2} 36 + log_{2} 5 b) (1/2) log x + log y  3 log z.
Solution:
We apply the log rules to compress each of the given expressions into a single logarithm.
a) log_{2} 36 + log_{2} 5
= log_{2 }(36 × 5) (∵ log_{b} m + log_{b} n = log_{b} mn)
= log_{2} 180
b) (1/2) log x + log y  3 log z
= [log x^{1/2} + log y]  log z^{3} (∵ n log m = log m^{n})
= log x^{1/2}y  log z^{3} (∵ log_{b} m + log_{b} n = log_{b} mn)
= log (x^{1/2}y/z^{3}) (∵ log_{b} m  log_{b} n = log_{b} m/n)
Answer: a) log_{2} 180 b) log (x^{1/2}y/z^{3}).

Example 2: If p = ln 2 and q = ln 6 then express ln 72 in terms of p and q.
Solution:
We have 72 = 36 × 2 = 6^{2} × 2. So
ln 72 = ln (6^{2} × 2)
By using natural logarithm rules,
ln 72 = ln 6^{2} + ln 2 (∵ ln mn = ln m + ln n)
= 2 ln 6 + ln 2 (∵ ln m^{n} = n ln m)
= 2q + p (∵ ln 6 = q and ln 2 = p)
Answer: 2q + p.

Example 3: Simplify 49^{log7 3} without logs.
Solution:
We will apply log rules to simplify the given expression.
49^{log73} = 7^{2}^{log73} (∵ 7^{2} = 49)
= 7^{log7 9} (∵ n log m = log m^{n} and 3^{2} = 9)
= 9 (∵ b^{logbx}_{ = x)}
Answer: 9.
FAQs on Log Rules
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  log_{b} 1 = 0 
Log of a Number With Same Base  log_{b} b = 1 
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 of Logarithm  log_{b} m^{n} = n log_{b} m 
Change of Base Rule  log_{b} a = (log a) / ( log b) 
Number Raised to Log  b^{logbx} = 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: log_{2} (xyz) = log_{2} x + log_{2} y + log_{2} 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').
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