Log Calculator — Free Online Logarithm Calculator

How to Use the Log Calculator

1. Select the logarithm base: Choose from four options in the dropdown: Common Log (base 10) for general-purpose calculations, Natural Log (base e) for calculus and continuous growth, Binary Log (base 2) for computer science, or Custom Base for any other base you need. Selecting Custom Base reveals an additional input field.
2. Enter the value (x): Type the positive number whose logarithm you want to compute. This must be greater than zero — logarithms of zero and negative numbers are undefined in real number mathematics. You can enter any positive decimal number.
3. Review the results: The result panel instantly displays the computed logarithm value to 10 decimal places, the formula showing your calculation, a verification statement (base raised to the result should equal your input), and comparison values showing the same number's logarithm in all three standard bases for reference.

The calculator also displays the antilog — the base raised to the power of the result — which should match your original input value. This verification feature helps confirm the answer and builds understanding of the inverse relationship between logarithms and exponentiation.

Logarithm Formulas and Properties

Variables Explained

• b (Base): The base of the logarithm. Must be positive and not equal to 1. Common choices are 10 (common log), e = 2.71828 (natural log), and 2 (binary log). The base determines the scale of the logarithm.
• x (Value/Argument): The positive number you are taking the logarithm of. Also called the argument of the logarithm. Must be strictly greater than zero for real-valued results.
• y (Result): The computed logarithm, representing the power to which the base must be raised to produce x. Can be positive, negative, or zero (log_b(1) = 0 for any valid base).

Step-by-Step Example

Calculate log_10(5000):

1. Express 5000 as a product: 5000 = 5 x 1000 = 5 x 10^3
2. Apply the product rule: log_10(5000) = log_10(5) + log_10(10^3)
3. Simplify: log_10(5) = 0.6990, and log_10(10^3) = 3
4. Add: 0.6990 + 3 = 3.6990

Verification: 10^3.6990 = 5000 (approximately). This confirms our answer. The result tells us that 10 must be raised to approximately the 3.699th power to equal 5000.

Practical Examples

Example 1: Rachel Measures Sound Intensity in Decibels

Rachel, an acoustics engineer, needs to calculate the decibel level of a sound with intensity 0.001 W/m^2, using the reference intensity of 10^(-12) W/m^2. The decibel formula uses base-10 logarithms: dB = 10 x log_10(I/I_0):

• Intensity ratio: 0.001 / 10^(-12) = 10^9
• log_10(10^9) = 9
• dB = 10 x 9 = 90 decibels

The sound measures 90 dB, equivalent to a lawn mower or motorcycle at 25 feet. This logarithmic scale means that a 10 dB increase represents a 10-fold increase in sound intensity, which is perceived by humans as roughly a doubling of loudness.

Example 2: Tom Calculates Investment Doubling Time

Tom wants to know exactly how many years it takes for an investment to double at 6% annual compound interest. The doubling time formula uses natural logarithms: t = ln(2) / ln(1 + r):

• r = 0.06 (6% as decimal)
• ln(2) = 0.6931
• ln(1.06) = 0.0583
• t = 0.6931 / 0.0583 = 11.90 years

Tom's investment doubles in approximately 11.9 years at 6% annual return. Compare this to the Rule of 72 estimate: 72/6 = 12 years — very close to the exact answer. For detailed investment projections, try our compound interest calculator.

Example 3: Lisa Analyzes Algorithm Complexity

Lisa is a software developer analyzing a binary search algorithm on a database of 1 million records. Binary search has O(log_2 n) time complexity, so she wants to know the maximum number of comparisons needed:

• n = 1,000,000 records
• log_2(1,000,000) = 19.93
• Maximum comparisons: 20 (rounded up)

Binary search can find any record among 1 million entries in at most 20 comparisons. This is the power of logarithmic complexity — doubling the dataset only adds one more comparison. A linear search would require up to 1 million comparisons by contrast.

Example 4: Daniel Measures pH

Daniel, a chemistry student, has a solution with a hydrogen ion concentration of 3.2 x 10^(-5) mol/L and needs to calculate its pH. The pH formula is: pH = -log_10([H+]):

• [H+] = 3.2 x 10^(-5)
• log_10(3.2 x 10^(-5)) = log_10(3.2) + log_10(10^(-5)) = 0.505 + (-5) = -4.495
• pH = -(-4.495) = 4.495

The solution has a pH of approximately 4.5, making it mildly acidic (similar to tomato juice). The logarithmic pH scale means a one-unit decrease in pH represents a 10-fold increase in hydrogen ion concentration.

Common Logarithm Values Reference Table

Tips and Complete Guide

Logarithmic Scales in Science

Many scientific measurements use logarithmic scales because the quantities they measure span enormous ranges. The Richter scale measures earthquake magnitude logarithmically — a magnitude 7 earthquake releases about 31.6 times more energy than a magnitude 6. The decibel scale measures sound intensity logarithmically because human hearing spans a trillion-fold range from the quietest to loudest sounds. The pH scale compresses hydrogen ion concentrations from 10^(-14) to 10^0 into a manageable 0-14 range. Without logarithms, these scales would be impractical to use.

Logarithms in Information Theory

Claude Shannon's information theory defines the bit as the fundamental unit of information using base-2 logarithms. The information content of an event with probability p is -log_2(p) bits. A fair coin flip (p = 0.5) carries -log_2(0.5) = 1 bit of information. A roll of a fair die (p = 1/6) carries -log_2(1/6) = 2.585 bits. Entropy, the average information content, is the basis for data compression algorithms. Understanding log_2 is essential for anyone working in data science, telecommunications, or cryptography.

The History of Logarithms

John Napier invented logarithms in 1614, calling them "artificial numbers" that converted multiplication into addition. Before electronic calculators, logarithm tables and slide rules were the primary computational tools for scientists, engineers, and navigators. Henry Briggs later developed common (base 10) logarithms, and the Euler number e (approximately 2.71828) gave rise to natural logarithms. The slide rule, which is essentially a physical logarithm table, was used by NASA engineers to design the Apollo spacecraft. Logarithms revolutionized calculation and remained the dominant computational method for over 350 years.

Connecting Logarithms to Exponential Growth

Logarithms are essential for solving exponential equations where the unknown is in the exponent. If you know that 1.05^t = 2 (when does a 5% annual growth lead to doubling?), take the logarithm of both sides: t x ln(1.05) = ln(2), so t = ln(2)/ln(1.05) = 0.6931/0.04879 = 14.21 years. This technique applies to compound interest, population growth, radioactive decay, and any exponential model. Logarithms turn the question "how long until this doubles?" into simple division.

Common Mistakes to Avoid

• Confusing log(a + b) with log(a) + log(b): The logarithm of a sum is NOT the sum of logarithms. log(a + b) has no simple expansion. The product rule states log(a x b) = log(a) + log(b), which applies to multiplication, not addition.
• Taking the log of zero or negative numbers: log(0) is undefined (approaches negative infinity), and log of negative numbers is not real. Always check that your argument is positive before computing logarithms.
• Mixing up bases: log_10 and ln give different results for the same input. log_10(100) = 2, but ln(100) = 4.605. Always specify which base you are using. If no base is written, convention varies: in pure math, log often means ln; in engineering and applied sciences, log usually means log_10.
• Forgetting that log_b(1) = 0 for any base: This follows from b^0 = 1. It is the logarithmic equivalent of the zero exponent rule and a useful quick check for your calculations.
• Incorrectly applying the power rule: log(x^n) = n x log(x) is correct, but log(n^x) = x x log(n), not n x log(x). The variable in the exponent and the base of the power behave differently. Pay attention to which quantity is the base and which is the exponent.

Frequently Asked Questions