Log Rules

Logarithm Rules or Log Rules are critical for simplifying complicated formulations that include logarithmic functions. Log Rules make it easier to calculate and manipulate logarithms in a variety of mathematical and scientific applications. Out of all these log rules, three of the most common are product rule, quotient rule, and power rule. Other than these, we have many rules of the logarithm, which we will discuss further in the article. This article explores, all the rules for logs including derivative and integral, in detail with the Logarithm Rules Examples. So, let’s start learning about all the rules logarithms have.

What are Log Rules?

Logarithm Rules in maths are the rules and laws that is used in simplification and manipulation of logarithmic function expressions. These principles create relationships between exponential and logarithmic forms and give a systematic technique to handle complicated logarithmic computations.

The key rules are as follows: product rule: which allows us to divide a product within a logarithm into a sum of separate logarithms; quotient rule: which allows us to divide a quotient within a logarithm into a difference of logarithms; power rule: which allows us to extract exponents from within a logarithm; base switch rule or change of base rule: which allows us to change the base of a logarithm.

These laws are crucial in many mathematical and scientific applications, making logarithms a valuable tool for solving equations, modelling exponential growth, and analysing large amounts of data.

Types of Logarithm

We usually deal with two kinds of logarithms:

• Common logarithm
• Natural logarithm

Note: There can be a logarithm with any real number as its base but these two i.e., common and natural logarithm, are the most common and standard ones.

Let’s discuss these types in detail.

Common Logarithm

A common logarithm, often known as log base 10 or simply log, is a mathematical function that represents the exponent to which a given number must be increased in order to reach a given number. It calculates the power of ten necessary to get a certain number.

For example, log₁₀(100) equals 2, because 10 raised to the power of 2 equals 100. The common logarithm of 100 in this case is 2, showing that 10² = 100. Common logarithms are used in many sectors, including science, engineering, and finance, to simplify huge number representations and help in computations requiring powers of  10.

Natural Logarithm

The natural logarithm is a mathematical function that expresses the logarithm to the base ‘e’ (Euler’s number, roughly 2.71828). It is the inverse of the exponential function and represents the amount of time required for a quantity to increase or decrease by a constant factor.

For example, ln (10) ≈ 2.30259 means that e multiplied by 2.30259 equals 10. The natural logarithm is used in many domains, including mathematics, physics, and finance, to describe phenomena that display exponential growth or decay, such as population expansion, radioactive decay, and compound interest computations.

What are Logarithm Rules?

Logarithmic operations can be conducted according to specific rules. These rules are known as:

• Product Rule
• Quotient Rule
• Zero Rule
• Identity Rule
• Power Rule or Exponential Rule
• Change of Base Rule
• Reciprocal Rule

Other then these common rules, we can also have some uncommon rules, such as:

• Logarithm Inverse Property
• Derivative of Log
• Integration of Log

Product Rule of Log

According to the product rule, the logarithm of a product is the sum of the logarithms of its elements.

Formula: log_a(XY) = log_aX + log_aY

Example: log₂(3 × 5) = log₂(3) + log₂(5)

Quotient Rule of Log

The quotient rule asserts that the logarithm of a quotient equals the difference of the numerator and denominator logarithms.

Formula: log_a(X/Y) = log_aX – log_aY

Example: log₃(9 / 3) = log₃(9) – log₃(3)

Zero Rule of Log

According to the zero rule, the logarithm of 1 to any base is always 0.

Formula: log_a(1) = 0

Example: log₄(1) = 0

Identity Rule of Log

According to the identity rule, the logarithm of a base to itself is always 1.

Formula: log_a(a) = 1

Example: log₇(7) = 1

Reciprocal Rule

According to the reciprocal rule of logarithms, the logarithm of a number’s reciprocal (1 divided by that number) is equal to the negative of the logarithm of the original number. In mathematical notation:

​Formula: log_a(1/X) = – log_a(X)

Example: log_a(1/2) = – log_a(2)

Power Rule or Exponential Rule of Log

According to the power rule, the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the base.

Formula: log_a(Xⁿ) = n × log_aX

Example: log₅(9²) = 2 × log₅(9)

Change of Base Rule of Log

The change of base rule enables you to calculate the logarithm of a number in a different base by employing a common logarithm (typically base 10 or base e). Change of Base Rule is also called Base Switch Rule.

Formula: log_a(X) = logᵦ(X) / logᵦ(a)

Example: log₃(7) = log₁₀(7) / log₁₀(3)

Logarithm Inverse Property

The logarithm inverse property asserts that calculating the logarithm of an exponentiated value yields the original exponent.

Formula: log_a(aⁿ) = n

Example: log₄(4²) = 2

Derivative of Log

The derivative of a function’s natural logarithm is the reciprocal of the function multiplied by the derivative of the function.

Formula: d/dx [ln(f(x))] = f'(x) / f(x)

Example: If y = ln(x²), then dy/dx = 2x / x² = 2/x

Integration of Log

Other than differentiation, we can also calculate the integral of the logarithm. The integral of Log function is given as follows:

Formula: ∫ln(x) dx = x · ln(x) – x + C = x · (ln(x) – 1) + C

Natural Log Rules

As natural and common both logs have only a difference of base, thus the rules for natural logs are the same as common log, which are already discussed. The only difference is that in natural log rules, instead of log (symbol of common log with base 10) we use ln (symbol for natural log base e). These rules can be stated 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
• e^{ln x} = x

Applications of Logarithm

Let us look at some of the applications of log.

• We utilise logarithms to calculate the acidity and alkalinity of chemical solutions.
• The Richter scale is used to calculate earthquake intensity.
• The amount of noise is measured in decibels (dB) on a logarithmic scale.
• Logarithms are used to analyse exponential processes such as the decay of ratio active isotopes, bacterial development, the spread of an epidemic in a population, and the cooling of a dead corpse.
• A logarithm is used to compute the repayment time of a loan.
• The logarithm is used in calculus to differentiate difficult equations and calculate the area under curves.

Product Rule of Logarithms

According to the product rule for logarithms, the logarithm of a multiplication of two terms is the same as the addition of the logarithms of those individual terms. In other words, this rule is expressed as log_b(mn) = log_b(m) + log_b(n). Let’s proceed to derive this rule.

Derivation Process:

Let’s start by assuming log_b(m) = x and log_b(n) = y. Converting both to their exponential forms, we obtain:

log_b(m) = x implies m = b^x … (1)

log_b(n) = y implies n = b^y … (2)

When we multiply equations (1) and (2) together,

mn = b^{x .} b^y

Utilizing the rules for multiplying exponents,

mn = b^{x + y}

Converting back into logarithmic form yields,

log_b(mn) = x + y

By substituting back for x and y,

log_b(mn) = log_b(m) + log_b(n)

Thus, we have derived the product rule of logarithms. This rule can be utilized in various ways, such as:

log(3a) = log 3 + log a log 10 = log(5×2) = log 5 + log 2 log3(ab) = log3 a + log3 b It’s important to note that the product rule for logarithms does not apply to log(m + n), which cannot be split into separate logarithms. This rule strictly pertains to the logarithm of a product, log(mn).

Logarithm Power Rule

The logarithm power rule states that when a logarithm’s argument is raised to a power, that exponent can be moved to the front of the logarithm. In other words, logb mn = n logb m. Let’s explore the derivation of this rule.

Derivation Process:

Start by assuming log_b m equals x. Converting this to its exponential form gives us:

b^x = m

Then, raise both sides to the power of n, resulting in:

(b^x)ⁿ = mⁿ

Applying the exponent power rule yields:

b^nx = mⁿ

Converting back to logarithmic form, we get:

log_b mⁿ = nx

By replacing x with log_b m, we arrive at:

log_b mⁿ = n log_b m

This concludes the derivation of the logarithm power rule. Below are several examples of how this rule is applied:

log 3z = z log 3 log y2 = 2 log y log3 yx = x log3 y

Quotient Rule of Logarithms

According to the quotient rule for logarithms, the logarithm of a division between two numbers is the subtraction of the logarithms of each number.

Specifically, the rule states that log_b (m/n) = log_b m – log_b n. Let’s proceed to derive this rule.

Derivation Process:

Suppose log_b m equals x and log_b n equals y. We’ll express these in their exponential forms.

log_b m = x implies m = b^x … (1)

log_b n = y implies n = b^y … (2)

When we divide equation (1) by equation (2),

m/n = b^x / b^y

Applying the quotient rule for exponents,

m/n = b^{x – y}

Converting back into logarithmic form,

log_b (m/n) = x – y

By substituting back for x and y,

log_b (m/n) = log_b m – log_b n

Thus, we have derived the quotient rule for logarithms. This rule can be utilized as follows:

log (y/3) = log y – log 3

log 25 = log (125/5) = log 125 – log 5

log7 (a/b) = log7 a – log7 b

It’s important to note that the quotient rule does not imply anything for log (m – n).

Related Topics:

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

Solved examples of Log Rules

Example 1: Simplify log₂(4 × 8).

Solution:

Using the product rule, we split the product into a sum of logarithms:

log₂(4 × 8) = log₂(4) + log₂(8) = 2 + 3 = 5.

Example 2: Simplify log₄(16 / 2).

Solution:

Using the quotient rule, we divide the quotient into a difference of logarithms:

log₄(16 / 2) = log₄(16) – log₄(2) = 2 – 0.5 = 1.5.

Example 3: Simplify log₅(25³).

Solution:

Using the power rule, we can bring down the exponent as a coefficient:

log₅(25³) = 3 × log₅(25) = 3 × 2 = 6.

Example 4: Convert log₃(7) into an expression with base 10.

Solution:

Using the base switch rule, we divide by the logarithm of the new base:

log₃(7) = log₁₀(7) / log₁₀(3) ≈ 1.7712

Example 5: Evaluate log₇(49) using the change of base rule with base 2.

Solution:

Using the change of base rule with base 2:

log₇(49) = log₂(49) / log₂(7) = 5 / 1.807 = 2.77 (approx).

Practise Questions on Log Rules

Problem 1: Simplify the expression: log₂ (4) + log₂ (8).

Problem 2: Simplify: log₅ (25) – log₅ (5).

Problem 3:  Simplify the expression: log₃ (9²).

Problem 4: Express log₄ (25) in terms of common logarithms.

Problem 5: Simplify using Log Rules: log₇ (49) + 2  log₇ (3).

Problem 6: Solve for x: log₂ (x) = 3.

Problem 7: Solve for x: 2^{3x – 1} = 8.

Log Rules – FAQs

What are Logarithm Rules?

Logarithm rules are a collection of recommendations for manipulating and simplifying formulas using logarithmic functions. They offer a systematic method for dealing with complicated computations and interactions between exponentials and logarithms.

How many Key Logarithm Rules are There?

The product rule, quotient rule, power rule, base switch rule, and change of base rule are all major logarithm rules. These principles permit logarithmic expression modifications and computations.

What is Logarithmic Product Rule?

According to the product rule, the logarithm of a product is equal to the sum of the logarithms of the individual factors: logₐ(xy) = logₐx + logₐy.

What are Two Types of Logarithms?

The two most commonly used logarithm types are:

• Common Logarithm or Base 10 Logarithm
• Natural Logarithm or Base e Logarithm

What is Log Rule for Change of Base?

According to change of base rule of log, log_a(b)=[log_c(b)]/[log_c(a)], where c is any positive real number.

What is Log 0?

The logarithm of zero is unknown. We never acquire the number 0 by raising any value to the power of any other value.

What is Log 1?

Because of the zero rule, the logarithm of 1 to any base is always 0 i.e., log_a(1) = 0.

What is Logarithm of any Number to Itself as Base?

According to the identity rule, the logarithm of a base to itself is always 1 i.e., log_a(a) = 1.

What is the Relationship between Logarithms and Exponentials?

Logarithms and exponentials are inverse operations. A logarithm tells you the exponent needed to reach a certain number, while an exponential raises a base to an exponent.

What are the 7 Rules of Logarithms?

The 7 Rules of Logarithms include

• Product Rule
• Quotient Rule
• Power Rule
• Change of Base Rules
• Zero Rule
• Identity Rule
• Negative Rule

These rules are used for simplifying logarithmic expressions.

What is Log Exponent Rule?

Log exponent rule states that log base b of a^x is equal to x times log base b of a i.e., log_ba^x = x log_ba.

What is the Key Difference between Common Log and Natural Log?

The key difference between common and natural log is that common logs use base 10, while natural logs use the mathematical constant ‘e’ as their base.

What is the Derivative Rule for Log?

The derivative rule for log functions is: d/dx[log_b(x)] = 1 / (x ln(b)), where ‘b’ is the base of the logarithm.

What is Base Switch Rule?

According to the Base Switch Rule, the base of any logarithm can be changed to any other desired base using the formula: loga(X) = logb(X) / logb(a).