Enter any two values to calculate the third in the equation logb(x) = y. Use "e" as base for natural log.
logb(x) = y
Base (b)or "e" for natural log
Argument (x)
Result (y)
About This Log Calculator
This Logarithm Calculator solves the equation logb(x) = y by finding the missing value. Enter any two of the three variables — base, argument, or result — and the calculator instantly computes the third. Supports all positive bases (except 1) and natural logs with base "e". Perfect for students, engineers, and anyone working with exponential relationships.
How to Use
1. Understand the Equation — The calculator solves logb(x) = y, where b is the base, x is the argument, and y is the result.
2. Enter Any Two Values — Fill in the base, argument (x), or result (y). Leave the third field empty.
3. Use "e" for Natural Log — Type "e" in the base field to calculate natural logarithms (ln).
4. Click Calculate — Press the "Calculate" button to find the missing value.
5. Review the Result — The calculator shows the computed value and the verification equation by = x.
How It Works & Formula
A logarithm answers the question: "To what power must the base be raised to get the argument?" The equation logb(x) = y is equivalent to by = x.
logb(x) = y ⇔ by = x
Where:
• b = Base (must be > 0 and b ≠ 1)
• x = Argument (must be > 0)
• y = Logarithm result (any real number)
Natural Logarithm: ln(x) = loge(x)
Natural logs use Euler's number (e ≈ 2.71828) as the base and are fundamental in calculus and exponential growth models.
Change of Base Formula: logb(x) = logk(x) / logk(b)
This property allows conversion between any logarithm bases, useful when your calculator only has log10 or ln functions.
Mathematical Expressions
If logb(x) = y, then by = x
logb(x × y) = logb(x) + logb(y)
logb(x / y) = logb(x) − logb(y)
logb(xy) = y × logb(x)
Real-Life Examples
Example 1: Find the logarithm of 100 with base 10.
log10(100) = y
10y = 100
102 = 100
y = 2
log10(100) = 2 because 10² = 100.
Example 2: Find the argument x given log₂(x) = 5.
log₂(x) = 5
2⁵ = x
x = 32
Since 2 raised to the 5th power equals 32, the argument x is 32.
Example 3: Natural logarithm of e³.
ln(e³) = y
ey = e³
y = 3
Natural log of e³ equals 3 because ln(en) = n.
Tip: Logarithms are used in many real-world applications — earthquake magnitudes (Richter scale), sound intensity (decibels), pH levels in chemistry, and compound interest calculations.
Common Logarithm Values Reference
Quick reference table for frequently used logarithm values.
Exponential Form
Logarithmic Form
Result
10⁰ = 1
log₁₀(1)
0
10¹ = 10
log₁₀(10)
1
10² = 100
log₁₀(100)
2
10³ = 1,000
log₁₀(1,000)
3
10⁴ = 10,000
log₁₀(10,000)
4
e⁰ = 1
ln(1)
0
e¹ = e ≈ 2.718
ln(e)
1
2⁰ = 1
log₂(1)
0
2¹ = 2
log₂(2)
1
2² = 4
log₂(4)
2
2³ = 8
log₂(8)
3
2⁴ = 16
log₂(16)
4
2⁵ = 32
log₂(32)
5
2¹⁰ = 1,024
log₂(1,024)
10
Remember: logb(1) = 0 for any valid base, and logb(b) = 1.
Quick Logarithm References
Common logarithm values at a glance.
log(1) = 0
log(10) = 1
log(100) = 2
log(1000) = 3
ln(1) = 0
ln(e) = 1
ln(e²) = 2
log₂(2) = 1
log₂(4) = 2
log₂(8) = 3
log₂(16) = 4
log₂(32) = 5
log(0) = undefined
log(-1) = undefined
Frequently Asked Questions
What is a logarithm? +
A logarithm answers the question: "To what exponent must the base be raised to get the argument?" For example, log₁₀(100) = 2 because 10² = 100. Logarithms are the inverse operations of exponentiation.
What is the difference between log and ln? +
"log" typically means base-10 logarithm (common log) in many contexts, while "ln" means natural logarithm with base e (≈ 2.71828). In higher mathematics, "log" often refers to natural log. This calculator accepts both — specify "e" in the base field for natural log.
Why can't the base be 0, 1, or negative? +
Base 0 or 1 would make the logarithm undefined or ambiguous. Base 1 would mean 1 raised to any power equals 1, so log₁(x) would be undefined for x ≠ 1. Negative bases lead to complex numbers for fractional exponents. Valid bases are positive numbers not equal to 1.
Why must the argument x be positive? +
For real-valued logarithms, the argument must be positive because a positive base raised to any real power always yields a positive result. log(-1) is undefined in the real number system (it exists in complex analysis as iπ).
What is the change of base formula and why is it useful? +
The change of base formula states log_b(x) = log_k(x) / log_k(b). It allows you to compute logarithms with any base using a calculator that only has log₁₀ or ln functions. For example, log₂(32) = log₁₀(32) / log₁₀(2) = 1.5051 / 0.3010 = 5.
What are common real-world uses of logarithms? +
Logarithms appear in many fields: earthquake magnitude (Richter scale — logarithmic), sound intensity (decibels), pH levels in chemistry (logarithmic), compound interest calculations, population growth models, computer science (binary logarithms), and information theory (entropy).
• Understanding Exponential Functions – The inverse relationship between logs and exponents.
• Natural Logarithms in Calculus – Why ln is essential for derivatives and integrals.
• Logarithms in Computer Science – Binary logs and algorithm complexity (Big O notation).
• Historical Development of Logarithms – How Napier revolutionized calculation in the 1600s.