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Log Calculator (Logarithm) is not yet available in 한국어. Showing the English version below. All formulas and calculations work the same.
마지막 업데이트: 2026년 7월 31일
Log Calculator (Logarithm)
공식
log mode: result = log_b(x) = ln(x) / ln(b); antilog mode: result = b^y
여기서:
- b=Logarithm base, where b > 0 and b ≠ 1
- x=Input value for log mode
- y=Exponent used in antilog mode
- r=Primary result
- ln(x)=Natural logarithm of x
- log₁₀(x)=Common logarithm of x
- log₂(x)=Binary logarithm of x
풀이 예제
Common log example with base 10
Compute a familiar logarithm using a custom base.
- 1Validate the base: 10 is positive and not equal to 1
- 2Apply log₁₀(100)
- 3Use change of base: ln(100) / ln(10)
- 4Read the supporting ln, log₁₀, and log₂ values
Custom base logarithm
Find the exponent on 3 that produces 81.
- 1Recognize that 81 = 3^4
- 2Therefore log₃(81) = 4
- 3Cross-check with ln(81) / ln(3)
- 4Review the supporting logs for x
Antilog with base 10 and exponent 2
Use the inverse operation to recover the original value.
- 1Switch to antilog mode
- 2Use the formula b^y
- 3Compute 10^2 = 100
- 4No change-of-base step is needed in this mode
Invalid base equal to 1
A logarithm base cannot be 1 because powers of 1 never vary.
- 1Check the base requirements first
- 2A valid base must be positive and different from 1
- 3Base 1 makes the logarithm undefined
- 4Choose another base and recalculate
소개
The Log Calculator (Logarithm) handles both custom-base logarithms and inverse antilog calculations in one place. In log mode it computes log_b(x) and also shows ln(x), log₁₀(x), and log₂(x) for the same input value. It also explains the change-of-base idea, making the calculator useful for algebra, science classes, coding, and engineering work.
Understanding Custom-Base Logs
A logarithm with base b asks which exponent on b produces x. This general form includes common logarithms, natural logarithms, binary logarithms, and any other valid positive base except 1.
log_b(x) asks for an exponent
The base must be positive
The base cannot equal 1
The input x must be positive in log mode
Different bases describe the same relationship in different scales
Change of Base Formula
When a calculator or programming language does not offer a direct custom-base log, you can compute it with natural logs or common logs. That is why the formula log_b(x) = ln(x) / ln(b) is central to this tool.
Converts any valid base to a familiar one
Works with natural logs or common logs
Makes custom-base calculations easy to verify
Explains why supporting log outputs are useful
Builds confidence in manual cross-checks
If b = 10, then log_b(x) reduces to the common logarithm log₁₀(x).
Log Mode and Antilog Mode
The calculator supports both directions of the inverse relationship. Use log mode to find an exponent from a base and value, or use antilog mode to raise the base directly to the chosen exponent.
Log mode uses base and x
Antilog mode uses base and exponent
The result field is always the primary answer
Supporting log outputs apply to x in log mode
Antilog mode skips change-of-base math
How to Use This Calculator
Pick the mode first, then fill in the matching inputs. The calculator checks the domain automatically and returns both the main answer and supporting context for interpretation.
Choose log_b(x) or b^y mode
Enter a valid base b
Enter x for log mode or y for antilog mode
Read the result field first
Use the change-of-base information for verification
Supporting Log Values
In log mode, the calculator also returns ln(x), log₁₀(x), and log₂(x). These extra values help connect classroom notation with scientific calculators, spreadsheet functions, and programming libraries.
naturalLog gives ln(x)
log10 gives log₁₀(x)
log2 gives log₂(x)
All supporting logs use the same x input
Rounded outputs make comparison easier
Validation Rules
The main restrictions come from logarithm domains and base rules. The calculator enforces them before computation so you immediately know whether a result is defined in the real-number system.
Base b must be greater than 0
Base b must not equal 1
x must be greater than 0 in log mode
Exponent must be finite in antilog mode
Unsupported modes return an explicit error
A base of 1 fails because 1 raised to any power is still 1, so it cannot generate varying outputs.
Common Mistakes to Avoid
Many errors come from mixing up the base, the input value, and the exponent. Checking whether the final answer makes sense as an exponent or as a power often reveals the issue immediately.
Using base 1 by mistake
Entering a non-positive x in log mode
Confusing x with the exponent field
Assuming all logs are base 10
Ignoring the inverse relationship between log and antilog
자주 묻는 질문
What does log_b(x) mean?
It means the exponent you must apply to the base b in order to produce x.
Why must the base be positive and not equal to 1?
A valid logarithm base must be positive, and base 1 is not allowed because 1 raised to any exponent is always 1.
Why must x be positive in log mode?
Real logarithms are defined only for positive input values, so x must be greater than 0.
What is change of base?
It is the identity log_b(x) = ln(x) / ln(b), which lets you compute a custom-base logarithm using a more familiar logarithm function.
What does antilog mode calculate?
It computes b^y directly, which is the inverse operation of taking a logarithm with base b.
Why are ln(x), log₁₀(x), and log₂(x) shown?
They help compare the same x-value across common bases and make it easier to verify a custom-base logarithm.
Can the exponent be negative in antilog mode?
Yes. Negative exponents are valid and produce reciprocal powers such as 10^-2 = 0.01.
Who uses custom-base logarithms?
Students, engineers, scientists, programmers, and analysts use them in algebra, modeling, signal work, and code-related calculations.