Logarithmic Equation Solver

Logarithmic Equation Calculator

Solve equations of the form:

logb(x) = c

Enter values below:

Base (b): Result (c):

Solution will appear here.

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Logarithmic Equations – Complete Mathematical Explanation

Logarithmic equations are equations that involve logarithms and unknown variables. They are deeply connected to exponential equations and play a crucial role in algebra, science, engineering, finance, and computer science.

In this article, we will explore logarithmic equations from the very beginning. You will understand what logarithms are, why they exist, how logarithmic equations work, and how to solve them step by step with confidence.

1. What Is a Logarithm?

A logarithm answers the question:

"To what power must a base be raised to obtain a given number?"

For example:

log10(100) = 2

This means:

10² = 100

Logarithms are the inverse operation of exponentiation.

2. Logarithmic and Exponential Forms

Every logarithmic equation can be rewritten as an exponential equation.

logb(x) = y  ⇔  by = x

This equivalence is the foundation of solving logarithmic equations.

3. Why Logarithms Exist

Logarithms were invented to simplify complex calculations involving large numbers. Before calculators, logarithms transformed multiplication into addition and division into subtraction.

Today, logarithms are used to:

• Measure sound intensity (decibels)
• Measure earthquakes (Richter scale)
• Analyze algorithms
• Model exponential growth and decay

4. Types of Logarithmic Equations

Type 1: Simple Logarithmic Equation

log2(x) = 3

Rewrite in exponential form:

2³ = x
x = 8

Type 2: Logarithm with Coefficient

2 log(x) = 4

Divide both sides by 2:

log(x) = 2
x = 100

5. Domain Restrictions

One of the most important rules of logarithms is:

The argument of a logarithm must be positive.

That means:

x > 0

Any solution that makes the argument zero or negative is invalid and must be rejected.

6. Solving Logarithmic Equations – Step by Step

1. Isolate the logarithm
2. Rewrite in exponential form
3. Solve for the variable
4. Check domain restrictions

7. Worked Example 1

log5(x) = 2

Rewrite:

5² = x
x = 25

Check:

log5(25) = 2 ✓

8. Worked Example 2

log(x − 1) = 2

Rewrite:

x − 1 = 100
x = 101

Domain check:

x − 1 > 0 → x > 1 ✓

9. Common Logarithms

A logarithm without a written base usually means base 10:

log(x) = log10(x)

10. Natural Logarithms

The natural logarithm uses base e:

ln(x) = loge(x)

Natural logarithms are essential in calculus, physics, and growth models.

11. Logarithmic Properties

• log(ab) = log(a) + log(b)
• log(a/b) = log(a) − log(b)
• log(aⁿ) = n·log(a)

12. Solving Multi-Log Equations

log(x) + log(x − 1) = 2

Combine:

log[x(x − 1)] = 2

Rewrite:

x(x − 1) = 100
x² − x − 100 = 0

13. Checking for Extraneous Solutions

Logarithmic equations often produce solutions that are algebraically correct but mathematically invalid due to domain restrictions.

14. Real-World Applications

• Sound intensity
• pH scale
• Earthquake magnitude
• Computer science algorithms
• Finance and economics

15. Common Mistakes

• Ignoring domain restrictions
• Forgetting base rules
• Mixing logarithmic laws incorrectly
• Not checking final answers

16. Final Thoughts

Logarithmic equations are powerful tools that reverse exponential growth. Once you understand how to convert between logarithmic and exponential forms, solving them becomes logical and systematic.

Use the calculator above to verify your results and strengthen your intuition.