Log Rules: The Crucial Mathematical Rules to Know

If you are someone who is taking a college or high school mathematics class, you are most likely to cover the logarithms section and rules of Logarithm. But what is Logarithm? Why does this letter “e” pop up in this mathematical expression? Natural logs might seem a bit difficult. However, once you get a glimpse of the Log rules, you can easily solve the complicated logarithm problems. Here, we explain 7 important rules for Logarithm or laws of Logarithm.

What is "ln"?

The Logarithm or ln can be defined as the inverse for “e”. Here, this letter “e” represents the mathematical constant known as the natural exponent. “e” comes as a mathematical constant with a properly set value. The value carried by “e” equals 2.71828. “e” appears at several mathematical instances for Logarithm rules which include scenarios such as growth equations, compound interest, as well as decay equations. ln(x) is termed as the time required for grow into “x”. On the other hand, ex is termed as the growth which has occurred post the time “x”.

The Rules for Logarithm

There are 7 primary rules that are required to be adhered to when working upon the Log properties.
Here are some ways on how to expand Logarithm.

Product Rule

ln(x)(y) = ln(x) + ln(y)
The log result for multiplication of the variables x & y is actually the sum of ln(x) and ln(y)
For example: ln(8)(6)= ln(8) + ln (6)

Quotient Rule

ln(x/y)= ln(x) – ln(y)
The log result for the division of the variables x & y is actually the difference of in(x) and ln(y)
For example: ln(7/4)= ln(7) – ln(4)

Zero Rule

ln(x)=0
The log result, in this case, is equal to zero
For example: ln(1)=0

Idenity Rule

Inx(x)=1
In this case, Logarithm of the number which is equal with its base is equal to 1
For example: ln₃(3)=1

Reciprocal Rule

ln(1/x)= -ln(x)
The log result for the reciprocal result of x here is the exact opposite of ln (x)
For example: ln(1/3)= -ln(3)

Power Rule

ln(xy) = y* ln(x)
The log result here for x is raised to power by y around y times for ln(x)
For example: ln(52)= 2* ln(5)

Log of Exponent Rule

lnx(xk)= k
In this case the Logarithm for exponential number with the same base as log, the result is the exponent
For example: ln₃(32)=2

Key Properties of Logarithm

Apart from these 4 rules discussed here, there are many properties for Logarithm you need to know. Here are some popular log properties.

Scenario	(ln) Property
(ln) of any negative number	The (ln) for any negative number = undefined
(ln) of 0	ln(0) = undefined
(ln) of 1	ln(1) = 0
ln of Infinity	ln(∞) = ∞
(ln) of e	ln(e) = 1
(ln) of e to the power of x	ln(ex) = x
e to the power of ln	Eln(x)=x

Example-1:

Given below is a mathematical expression, evaluate the same using Log rules.
Log₂4+Log₂8.
Here, the numbers 4 and 8 are the exponential numbers while the base is equal as 2. Here the power rule is applied to the expression followed by the identity rule. Doing this will provide the final result which has been explained below:

Log₂4+Log₂8
=Log₂2²+Log₂2³
=2 Log₂2+3Log₂2
=2(1) +3(1)
=2+3
Therefore, Log₂8+Log₂4=5

Example-2:

Given below is a mathematical expression, solve the same using the Log rules.
Log₃162-Log₃2
In this case, the number 162 cannot be expressed as the exponential number that comes with the base 3. Here, no straight rules are applicable in the direct manner. However, logarithm rules in this case can actually be applied in the reverse manner! Take note that these log expressions can easily be expressed as a single logarithm term via use of quotient rule in the backward manner.

Log₃162-Log₃2
=Log₃ {162/2}
=Log₃ (81)
=Log₃34
=4 Log₃3
=4 (1)
Log₃162- Log₃2= 4

Example-3:

Given below is a mathematical expression that needs to be evaluated using the Log rules. Log₅500- 2Log₅2+Log₄32+Log₄8 By the looks of it, this expression suggests that there are multiple things occurring at the same time. The first move here is the simplification of each logarithm number. This will obviously begin with application of the log rules. When you observe the expression, there are two different bases involved: 4 and 5.
In this case, you need to put together the expressions with same base. Now, these bases with same quotient shall be simplified separately. For the ones with 5 as base, the Power Rule shall be applied first. After this, the Quotient Rule shall be applied. For the log with 4 as base, the Product Rule shall be applied immediately. This will lead to the final result by addition of the values obtained. So by applying the process explained above,

Log₅500-2Log₅2+Log₄32+Log₄8
=Log₅500-Log₅22+Log₄432+Log₄8
=Log₅500-Log₅4+Log₄32+Log₄8
=Log₅ (500/4) + Log₄ (32*8)
=Log₅ (125) + Log₄ (256)
=Log₅ (53) + Log₄ (44)
=3* Log₅5 + 4* Log₄ (4)
= 3(1) + 4(1)
=7

Example-4:

Given below is a mathematical expression, solve the same using logarithm rules:

Log₃(27x²y⁵)
Inside this parenthesis is the product of the factors. Here, you need to apply the Logarithm Product Rule and break them as the sum of separate log expressions. Try to simplify the numerical expressions to derive the exact value when possible. Use the Identity Rule to simplify this process further.

Log₃(27x²y⁵)

= Log₃(27) + Log₃(x²) +Log₃(y⁵)

= Log₃(33) + Log₃(x²) + Log₃(y⁵)

=3Log₃(3) + 2Log₃(x) + 5Log₃(y)

=3(1) +2 Log₃(x) + 5Log₃(y) {Answer}

Example-5:

Given below is a mathematical expression, decode the same using logarithm rules. Log₇{49m⁶/K³} The approach for this mathematical expression is to apply Logarithm quotient rule. After the simplification of the expression, the Product Rule needs to be applied to obtain the sum for logarithm expressions.

Log₇{49m⁶/K³}
=Log₇(47m⁶) – Log₇(K³)
= Log₇(49) + Log₇(m⁶) – Log₇(K³)
=Log₇(72) +Log₇(m⁶) –Log₇(K³)
=2Log₇(72) + 6Log₇(m) - 3Log₇(k)
=2(1) +6Log₇(m) - 3Log₇(k)
=2+ 6Log₇(m) - 3Log₇(k)