Logarithm Calculator

Solve for any variable in the equation y = log_b(x)

Examples

  • log₂(8) = 3 (Since 2³ = 8)
  • log₁₀(100) = 2 (Since 10² = 100)
  • Find x if log₃(x) = 4. (x = 3⁴ = 81)

Important Note

The logarithm log_b(x) = y is equivalent to the exponential equation b^y = x. The calculator can solve for y, x, or b.

Other Titles
Understanding Log Calculator (Logarithm): A Comprehensive Guide
Dive into the world of logarithms, the inverse operation to exponentiation, and their essential role in science, finance, and mathematics.

Understanding Logarithms: A Comprehensive Guide

A logarithm is the power to which a number (the base) must be raised to produce a given number. In simple terms, a logarithm answers the question: 'How many of one number do we multiply to get another number?'
The relationship is expressed as log_b(x) = y, which is equivalent to b^y = x. Here, 'b' is the base, 'x' is the argument (or number), and 'y' is the logarithm.
Common and Natural Logarithms
1. Common Logarithm (log): This is the logarithm with base 10. It's widely used in science and engineering. If no base is written, it's usually assumed to be 10 (e.g., log(1000) = 3).
2. Natural Logarithm (ln): This is the logarithm with base 'e' (Euler's number, approx. 2.71828). It is crucial in calculus, physics, and finance for modeling continuous growth.

Basic Examples

  • log₂(16) = 4 because 2⁴ = 16
  • log₅(25) = 2 because 5² = 25
  • ln(e) = 1 because e¹ = e
  • log(100) = 2 because 10² = 100

Step-by-Step Guide to Using the Log Calculator

1. Choose What to Solve For
Select from the dropdown menu whether you want to calculate the Logarithm (y), the Number (x), or the Base (b).
2. Enter the Known Values
Based on your selection, input the two known values. For instance, to find log₂(64), you would select 'Logarithm (y)', enter 2 for the Base (b) and 64 for the Number (x).
3. Calculate the Result
Click 'Calculate'. The calculator will provide the missing value. It enforces domain rules: the base 'b' must be positive and not 1, and the number 'x' must be positive.

Usage Examples

  • To find y in `log₃(81) = y`: Select 'Logarithm (y)', input b=3, x=81. Result: y=4.
  • To find x in `log₅(x) = 2`: Select 'Number (x)', input b=5, y=2. Result: x=25.
  • To find b in `log_b(49) = 2`: Select 'Base (b)', input x=49, y=2. Result: b=7.

Real-World Applications of Logarithm Calculations

Measuring Acidity (pH)
The pH scale is logarithmic (base 10), measuring the concentration of hydrogen ions in a solution. pH = -log[H⁺].
Earthquake Magnitude (Richter Scale)
The Richter scale measures earthquake intensity logarithmically. An increase of 1 on the scale corresponds to a 10-fold increase in amplitude.
Sound Intensity (Decibels)
The decibel (dB) scale for sound intensity is logarithmic. A 10 dB increase represents a 10-fold increase in sound intensity.
Finance (Compound Interest)
Logarithms are used to determine the time required for an investment to reach a certain value with continuous compounding.

Practical Examples

  • A solution with pH 4 is 10 times more acidic than one with pH 5.
  • A 7.0 magnitude earthquake is 100 times more intense than a 5.0 magnitude one.
  • A 70 dB sound (a vacuum cleaner) is 1,000 times more intense than a 40 dB sound (a quiet library).

Common Misconceptions and Correct Methods in Logarithms

Misconception 1: log(x+y) = log(x) + log(y)
  • Wrong: Treating the logarithm of a sum as the sum of logarithms.
  • Correct: The correct property is the product rule: log(xy) = log(x) + log(y). There is no simplification for log(x+y).
Misconception 2: log(x)/log(y) = log(x-y)
  • Wrong: Confusing the division of logs with the log of a difference.
  • Correct: The correct property is the quotient rule: log(x/y) = log(x) - log(y).
Misconception 3: (log(x))^n = n*log(x)
  • Wrong: Misapplying the power rule.
  • Correct: The power rule is log(x^n) = n*log(x). The term (log(x))^n cannot be simplified in the same way.

Correction Examples

  • log(10*100) = log(10) + log(100) = 1 + 2 = 3.
  • log(100/10) = log(100) - log(10) = 2 - 1 = 1.
  • log(10²) = 2*log(10) = 2*1 = 2.

Mathematical Derivation and Examples

The change of base formula is one of the most important properties of logarithms, allowing us to calculate a logarithm of any base using a calculator that only has common (base 10) and natural (base e) logs.
Change of Base Formula
The formula states: logb(x) = logc(x) / log_c(b), where 'c' can be any new base (typically 10 or e).
To derive this, let y = logb(x). This means b^y = x. Now, take the log base 'c' of both sides: logc(b^y) = logc(x). Using the power rule, we get y * logc(b) = logc(x). Finally, solve for y: y = logc(x) / log_c(b).

Change of Base Example

  • Calculate log₂(7): Using natural log (ln), log₂(7) = ln(7) / ln(2) ≈ 1.9459 / 0.6931 ≈ 2.807.
  • Calculate log₅(125): log₅(125) = log(125) / log(5) ≈ 2.0969 / 0.6989 ≈ 3.