Tips and Tricks to Logarithm Question

The logarithm is the inverse operation of exponentiation. It is defined as the power to which the base number must be raised to get the given number.

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

This article provides essential tips and tricks for tackling logarithm questions effectively, covering key concepts, formulas, and rules that every learner should grasp before diving into practice.

Types of Logarithms

There are two main types of logarithms you’ll encounter:

1) Common Logarithm: Written as log⁡(x), this log has a base of 10.

• It answers the question, “To what power must 10 be raised to equal x?”
• Common logs are often used in scales like pH, sound (decibels), and earthquake magnitudes.
• Example: log⁡(100) = 2 (because 10² = 100).

2) Natural Logarithm: Written as ln⁡(x), this log has a base of e (approximately 2.718).

• It answers the question, “To what power must e be raised to equal x?”
• Natural logs are used in natural growth and decay models, like population growth and radioactive decay.
• Example: ln⁡(e³) = 3.

Tips and Tricks for Logarithm

Below is a detailed table with rules, equations, samples, and quick tricks to solve logarithm questions. Remember the table below to quickly solve the questions.

Common Logarithmic Values for Base 10 (log⁡₁₀) Table

Tips for Solving Logarithmic Equations

Basic Logarithmic Equation

When you have a simple equation of the form log_⁡b(x) = y, you can rewrite it in its exponential form to solve for x: x = b^y

Example: Solve for x in log⁡₃(x) = 4.

Solution:

• Rewrite the equation in exponential form: x = 3⁴
• Calculate 3⁴: x = 81

So, x = 81

Equations with Logarithms on Both Sides

When you have an equation with logarithms on both sides and the same base, you can set the insides equal to each other.

Example: Solve for x in log₄(x + 3) = log₄(7).

Solution:

Set the insides equal to each other, since both sides have the same base (4), we can set the expressions inside the logarithms equal to each other:

• x + 3 = 7
• x = 4

so the value of x = 4.

Using Logarithmic Properties to Simplify

Sometimes, you’ll need to use logarithmic properties (product, quotient, or power rule) to simplify an equation before solving it.

Example: Solve for x in log⁡₂(4x) = 5

Solution:

Use the product rule to separate the terms:
log⁡₂(4) + log_⁡2(x) = 5

Simplify log⁡₂(4) (since 2² = 4, so log_⁡2(4) = 2):
2 + log_⁡2(x) = 5

Subtract 2 from both sides:
log_⁡2(x) = 3

Rewrite in exponential form: x = 2³ = 8
So, x = 8.

Handling Logarithmic Equations with Different Bases

If you encounter logarithmic equations with different bases, use the change of base formula to convert one or both sides to a common base.

Example: Solve log⁡₂(x) = log_⁡3(9)

Solution:

Convert log⁡₃(9), using change of base, the change of base formula states:

• log₃(9) = log₂(9)/log_⁡2(3)

Substitute into the Equation:

• log_2​(x) = log_2​(9)/log₂​(3)

Simplify the Right Side:

• ​Calculate log⁡₂(9) and log⁡₂(3) in terms of powers of 2:
• Since 9=3², we can use the power rule: log⁡₂(9) = log_⁡2(3²) = 2⋅log⁡₂(3).

Thus the equation becomes: log₂​(x) = 2⋅log₂​(3)​/log₂​(3)

• Cancel log⁡₂(3) on the Right Side: log_2​(x) = 2
• Rewrite in Exponential Form
• Convert log⁡₂(x) = 2 to exponential form: x = 2² = 4

So, x = 4

Solved Questions and Answers

Question 1: Solve an Exponential Equation Find the value of 'x' in the equation 3^x = 27.

Solution:

We can rewrite the equation using logarithms: x = log₃ (27).

Using the base-3 logarithm, we find x = 3, as 3³ = 27.

Question 2: Use logarithmic properties to evaluate log₂ (8) + log₂ (16).

Solution:

Using the property of the sum of logarithms, we have log₂ (8) + log₂ (16) = log₂ (8 * 16).

Simplifying further, we get log₂ (128).

Using the base-2 logarithm, log₂ (128) = 7, as 2⁷ = 128.

Question 3: Change of Base Calculation. Calculate log₅ (125) using the change of base formula.

Solution:

Using the change of base formula, we have log₅ (125) = log₁₀ (125) / log₁₀ (5).

Evaluating the logarithms, we get log₅ (125) ≈ 2.0969 and log₁₀(5) ≈ 0.69897

so, log₅ (125) = 2.0969 / 0.69897 ≈ 3

also, log₅ (125) = 3 , since 5³ = 125

Question 4: Solving Logarithmic Equations Solve the equation 2log₂(x) = 8.

Solution:

We can rewrite the equation as log₂(x) = 4.

Converting it to exponential form, we have x = 2⁴.

Simplifying, we get x = 16.

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• Logarithms
• Log Rules
• Log Table
• Logarithm Aptitude Questions and Answers
• Logarithm Quiz