Log Calculator

Compute common logarithms, natural logarithms, and custom-base logs with high precision, ideal for math, science, and engineering problems.

Common Log Base 10

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Natural Log Base e

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Log Custom Base

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Understanding Logarithms: Complete Guide with Properties, Examples, and Applications

Logarithms are inverse operations of exponentiation and are essential for working with ratios, growth rates, and very large or very small numbers in a compact way.

This guide mirrors the structure of your exponent tutorial: multiple boxes, rules, worked examples, applications, pitfalls, and practice problems, all tailored to base‑10, natural, and custom-base logs.

What Are Logarithms?

A logarithm answers the question: “To what power must the base be raised to obtain a given number?” In symbolic form, logb(x) is the exponent y such that by = x.

Key Components
  • Base: The number b being raised to a power (b>0 and b≠1).
  • Argument: The positive number x whose logarithm is taken.
  • Log value: The exponent y that satisfies by = x.
Example 1: Common log
log10(1000) = 3 because 103 = 1000.
Example 2: Natural log
ln(e2) = 2 because e2 is the number whose natural log equals 2.
Form Logarithmic Exponential Interpretation
Base 10 log10(100) 102 = 100 Answer is 2
Base e ln(e) e1 = e Answer is 1
Base 2 log2(8) 23 = 8 Answer is 3
Base 5 log5(1) 50 = 1 Answer is 0
Base 10 log10(1) 100 = 1 Answer is 0

The Core Laws of Logarithms

Log rules convert products, quotients, and powers into simpler sums and differences, making them crucial in algebra, calculus, and data analysis.

1. Product Rule

logb(MN) = logb(M) + logb(N)

Example:
log10(1000) = log10(10 × 100) = 1 + 2 = 3.
2. Quotient Rule

logb(M/N) = logb(M) − logb(N)

Example:
log10(100/10) = log10(100) − log10(10) = 2 − 1 = 1.
3. Power Rule

logb(Mk) = k·logb(M)

Example:
log10(1000) = log10(103) = 3·log10(10) = 3.
4. Change of Base Formula

logb(x) = loga(x) / loga(b) (often with a = 10 or a = e for calculators).

Example:
log2(8) = ln(8)/ln(2) = 3.
5. Log of 1

logb(1) = 0 for any valid base b because b0 = 1.

6. Log of the Base

logb(b) = 1 because b1 = b.

7. Domain Rule

For real logs, the base must be positive and not equal to 1, and the argument must be positive.

Law Formula Example Result
Product logb(MN) = logbM + logbN log10(2×5) log102 + log105
Quotient logb(M/N) = logbM − logbN log10(100/10) 2 − 1 = 1
Power logb(Mk) = k·logbM log10(104) 4
Base logb(b) = 1 ln(e) 1
One logb(1) = 0 log2(1) 0

Special Types of Logs Explained

In practice, three types of logs are used most often: base‑10 logs, natural logs (base e), and logs with arbitrary bases for discrete systems.

Common (base 10) logarithms

These are written as log(x) or log10(x) and are used heavily in engineering, decibels, and orders of magnitude.

Example 1: Sound level
If an intensity grows by a factor of 10, the decibel level increases by 10·log10(10) = 10 dB.
Natural logarithms (base e)

Natural logs ln(x) use base e and are fundamental in calculus, growth and decay models, and continuous compounding.

Example 2: Continuous compounding
For amount A = Pert, solving for t uses t = (1/r)·ln(A/P).
Logs with custom bases

Custom-base logs logb(x) appear in computer science, information theory, and any context with a natural base other than 10 or e.

Example 3: Binary entropy
Information content often uses log2, since each step doubles the number of states.
Notation Base Typical Use Example
log(x) 10 Engineering, decibels log(10,000) = 4
ln(x) e Calculus, growth ln(e3) = 3
log2(x) 2 Computer science log2(8) = 3
logb(x) b>0, b≠1 General models log3(9) = 2
loga(b) a Change of base log10(2) ≈ 0.3010

Step-by-Step Calculation Examples

These examples show how to evaluate logs, apply rules, and interpret calculator outputs step by step.

Example 1: Using the product rule

Simplify log10(2×50).

Solution:
Step 1: Apply product rule: log(2×50) = log(2) + log(50).

Step 2: Note that 50 = 5×10, so log(50) = log(5) + log(10) = log(5) + 1.

Step 3: Combine: log(2) + log(5) + 1 = log(10) + 1 = 1 + 1 = 2.

Step 4: Check: 2×50 = 100, and log(100) = 2.
Example 2: Solving for an exponent

Find x if 10x = 250.

Solution:
Step 1: Take log base 10: log(10x) = log(250).

Step 2: Use power rule: x·log(10) = log(250).

Step 3: Since log(10)=1, x = log(250) ≈ 2.3979.
Example 3: Using change of base

Evaluate log3(7) using natural logs.

Solution:
Step 1: Apply change of base: log3(7) = ln(7)/ln(3).

Step 2: Approximate ln(7) ≈ 1.9459, ln(3) ≈ 1.0986.

Step 3: Divide: 1.9459 / 1.0986 ≈ 1.770.

Step 4: Confirm using the “Log Custom Base” calculator.
Example 4: Continuous growth

Suppose a population follows P(t) = P0ekt. Find t when population doubles.

Solution:
Step 1: Set P(t) = 2P0 so 2P0 = P0ekt.

Step 2: Divide by P0: 2 = ekt.

Step 3: Take ln: ln(2) = kt, so t = ln(2)/k.

Step 4: ln(2) ≈ 0.6931 is a common constant in growth/decay problems.

Real-World Applications of Logarithms

Logs compress large numerical ranges and turn multiplicative relationships into additive ones, which is why they appear across science, engineering, and data analysis.

📈 Exponential Growth and Decay

Population, radioactive decay, and pharmacokinetics use logs to solve for time, rates, and half‑lives in exponential models.

🔊 Decibels in Audio

Sound intensity is measured on a log scale: dB = 10·log10(I/I0), so each power-of‑10 change is a fixed dB difference.

🌋 Richter Scale

Earthquake magnitudes use a logarithmic scale; an increase of 1 unit represents roughly 10 times more amplitude.

📊 Data Visualization

Log axes make it easier to see patterns when data spans several orders of magnitude, as in finance or astronomy.

💾 Algorithms and Complexity

Binary search, tree heights, and many algorithms have time complexity involving log2(n), linking logs to performance analysis.

🧪 Chemistry and pH

pH is defined as −log10[H⁺]; a one‑unit change corresponds to a tenfold change in hydrogen ion concentration.

Common Mistakes and How to Avoid Them

Many log errors come from ignoring domain restrictions or misusing log rules. Recognizing these patterns prevents incorrect results.

❌ MISTAKE 1: Taking logs of non‑positive numbers
log(0) and log(negative) are undefined in the real numbers.
✅ Always check that the argument is strictly positive before using a log.
❌ MISTAKE 2: Treating log(ab) like log(a)+b
Product rules apply inside the log, not outside.
✅ Only use log(ab) = log(a) + log(b) when both a and b are inside the log.
❌ MISTAKE 3: Forgetting the base in change‑of‑base
Using ln(x)/ln(2) when base is 10 leads to wrong answers.
✅ Match the base correctly: logb(x) = ln(x)/ln(b) or log(x)/log(b).
❌ MISTAKE 4: Ignoring restrictions on the base
Using base 1 or negative bases breaks log definitions.
✅ Ensure base b is positive and not equal to 1 before computing.
💡 Pro Tips for Success:
  • Rewrite log equations in exponential form to check answers.
  • Keep track of the base explicitly when switching between log and ln.
  • Use log rules (product, quotient, power) one at a time to avoid confusion.
  • Use the three calculators above to verify each algebraic step numerically.

Logarithms and Scientific Notation

Logs and scientific notation work together to describe huge and tiny values using powers of 10 instead of long strings of zeros.

Understanding logs of powers of 10

If a number is written as a × 10n with 1 ≤ a < 10, then log10(a × 10n) = log10(a) + n.

Example 1: Large number
3 × 108 has log10 value log(3) + 8 ≈ 0.4771 + 8 ≈ 8.4771.
Example 2: Small number
1 × 10−9 has log10 value −9.
Example 3: Back‑and‑forth
If log10(x) = 4.2, then x = 104.2 ≈ 1.58 × 104.
Standard Form Scientific Notation log10(x) Description
1,000,000 1 × 106 6 One million
0.001 1 × 10−3 −3 One thousandth
3,000,000,000 3 × 109 log(3) + 9 Three billion
0.00000052 5.2 × 10−7 log(5.2) − 7 Small measurement
1 1 × 100 0 Neutral reference
Converting with logs and powers of 10:
  1. Write x in scientific notation as a × 10n.
  2. Compute log10(a) with the log calculator.
  3. Add n to log10(a) to get log10(x).
  4. Reverse the process using 10log(x) to recover x.

Practice Problems with Solutions

Use these problems to practice log rules and calculator usage; open the solutions only after attempting each one.

Basic Level
Problem 1: Evaluate log10(1000)
Show Solution

103 = 1000, so log(1000) = 3

Problem 2: Evaluate ln(e4)
Show Solution

ln(e4) = 4·ln(e) = 4

Intermediate Level
Problem 3: Simplify log10(2×105)
Show Solution

log(2) + 5 ≈ 0.3010 + 5 = 5.3010

Problem 4: Compute log2(32) using change of base
Show Solution

log2(32) = ln(32)/ln(2) = 5

Advanced Level
Problem 5: Solve for x: ln(x) = 3
Show Solution

x = e3

Problem 6: Simplify log3(27) − log3(3)
Show Solution

log3(27) = 3, log3(3) = 1, so result = 3 − 1 = 2