Log Rules

Logarithm rules are used to simplify and work with logarithmic expressions. They help relate logarithms to exponents and make complex calculations easier.
A logarithm is the inverse of an exponent. It answers the question: "To what power must a base be raised to get a certain number?"

Out of all these log rules, three of the most common are product rule, quotient rule, and power rule.

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

List of Logarithm Rules

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 between 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). The change of Base Rule is also called the 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 the Log function is given as follows:

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

Note: As natural log and common both logs have only a difference of base, thus the rules for natural logs are the same as common log.

Also Read: Laws of Logarithms

Log Values from 1 to 10

The logarithmic values from 1 to 10 to the base 10 are:

Applications of Logarithms

Let us look at some of the applications of logs.

• We utilize 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 analyze exponential processes such as the decay of radioactive isotopes, bacterial development, the spread of an epidemic in a population, and the cooling of a 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.

Related Topics:

• Applications of logs
• Antilog Table
• Log Calculator
• 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).

Suggested Quiz

Edit Quiz

10 Questions

Solve for x: log⁡₂⁡(x−3)+log₂(x+5)=3

• A

  4

• B

  3.59

• C

  3

• D

  3.89

Explanation:

Combine logs: ⁡log₂((x−3)(x+5)) = 3

Rewrite in exponential form: (x−3)(x+5) = 2³ = 8

Expand and solve: x²+2x−15 = 8    
x²+2x−23 = 0

Using the quadratic formula:

[Tex]x = \frac{-2 \pm \sqrt{4 + 92}}{2} = \frac{-2 \pm \sqrt{96}}{2} = \frac{-2 \pm 4\sqrt{6}}{2} = -1 \pm 2\sqrt{6}[/Tex]​

Valid x: [Tex]x = -1 + 2\sqrt{6} \approx 3.898[/Tex]

If log_⁡ab=2 and log⁡_bc=3, find log_⁡ac.

• A

  5

• B

  6

• C

  3

• D

  10

Explanation:

Using the change of base formula:

log⁡_ac = log⁡_ab × log_⁡bc = 2×3 = 6

Simplify [Tex]\log \left(\frac{1000 \cdot \sqrt{10}}{10^3}\right)[/Tex]

• A

  1

• B

  0.5

• C

  0.3

• D

  0.7

Explanation:

Rewrite inside the log:
[Tex]\log \left(\frac{1000 \cdot \sqrt{10}}{1000}\right) = \log (\sqrt{10}) = \log (10^{1/2}) = \frac{1}{2} \log (10) = \frac{1}{2} \times 1 = 0.5[/Tex]

If log_⁡x 2 = a and log⁡_x 5 = b, express log⁡_x 20 in terms of a and b.

• A

  2a+b

• B

  a+2b

• C

  2a

• D

  2ab

Explanation:

log_x​20 = log_x​(2² × 5) = 2log_x​2 + log_x​5 = 2a+b

Evaluate: log_⁡4 8 + log_⁡8 4

• A

  11/6

• B

  12/6

• C

  13/6

• D

  10/6

Explanation:

Rewrite using change of base:

log⁡₄8+log_⁡84 = (log⁡ 8/log⁡ 4)+(log ⁡4/log ⁡8)

Since log ⁡8 = 3log ⁡2 and log ⁡4 = 2log ⁡2:

= 3/2+2/3=(9+4)/6=13/6.

If log_⁡x 4 = 2 and log_⁡x 8 = 3, find x.

• A

  4

• B

  8

• C

  3

• D

  2

Explanation:

Using properties of logarithms:
x² = 4 and x³ = 8
From x² = 4, we find x=2, which also satisfies x³ = 8
Therefore, x=2

Solve for x if log⁡₃ (x+1) − log_⁡3 (x−1) = 1

• A

  4

• B

  2

• C

  8

• D

  none of these

Explanation:

Combine logs:
log_⁡3 [(x+1)/(x−1)] = 1
Rewrite in exponential form:
(x+1)/(x−1) = 3
Cross-multiply and solve:
x+1 = 3(x−1)    
⇒ x+1 = 3x−3  
⇒ 4 = 2x  
⇒  x = 2

If log_⁡5 (3x−7) = 2, find x

• A

  10.80

• B

  10.55

• C

  10.67

• D

  10.79

Explanation:

Rewrite in exponential form:
3x−7 = 5² = 25
Solve for x:
3x=32
⇒ x = 32/3 ≈ 10.67

Solve the following logarithmic equation for x .

2log_ax = log_a18 + log_a(x-4).

• A

  6, 10

• B

  4, 7

• C

  12, 6

• D

  7, 6

Explanation:

As we know, log ( m × n ) = log m + log n

log_a(x²)= log_a(18 × (x-4))
x²= l18 × (x-4)
⇒ x² = 18x-72
⇒ x² - 18x + 72 = 0
⇒ (x - 6) (x - 12) =0
⇒ x = 6, 12.

If log_⁡3 x = 5, find log⁡₉ x

• A

  2.5

• B

  2.75

• C

  2.35

• D

  2.78

Explanation:

Rewrite log⁡₉ x using base change:

log_⁡9 x = log⁡₃ x/log_⁡3 9 = 5/2=2.5

Quiz Completed Successfully

Your Score :   2/10

Accuracy :  0%

Login to View Explanation

1/10 1/10 < Previous Next >