QuickCalculators solves logarithms for any of the three values in log_b(x) = y, covering common log, natural log, and custom bases, with log rules and change-of-base steps shown when useful. Enter the known parts, choose the base, and read the missing value with domain checks applied.
Solve a logarithm for any of its three values
A logarithm equation relates base, argument, and exponent: log_b(x) = y means b^y = x. Log Calculator rearranges for whichever quantity is unknown, provided the base and argument stay in the valid domain. Solving for the argument uses exponentiation; solving for the exponent uses the log definition directly; solving for the base uses roots or powers as needed.
Example: log base 2 of 32 equals 5 because 2^5 = 32. If the unknown is the argument in log_10(x) = 3, then x = 10^3 = 1000. QuickCalculators prints the matching exponential identity beside the numeric answer.
Read the logarithm as an exponent
Every logarithm answer is an exponent. Asking for log_b(x) asks what power of b produces x. That reading keeps log notation tied to exponentiation instead of treating "log" as an opaque button. The natural logarithm ln(x) is the special case with base e.
So log_10(100) = 2 because 10 squared is 100, and ln(e^3) = 3 because the base-e exponent is already visible. Inverse thinking helps: applying the matching exponential undoes the log.
Avoid this common mistake
Writing log(x) without a base does not mean the same thing everywhere. In many school and calculator contexts log means base 10, while in most programming languages and many advanced texts log means natural log base e. The base is not optional information. QuickCalculators requires an explicit base or a clearly labeled ln mode.
Ambiguous homework that says only "log" should be checked against the course convention before comparing answers. When in doubt, write log_10 or ln explicitly.
Apply the log rules
The product, quotient, and power rules rewrite logs of products, quotients, and powers as sums, differences, and scaled logs. Change of base converts any log into a ratio of logs in a convenient base. Log Calculator can apply those identities when simplifying or when evaluating an awkward base.
Rules in brief: log_b(mn) = log_b(m) + log_b(n), log_b(m/n) = log_b(m) - log_b(n), and log_b(m^k) = k · log_b(m). Change of base: log_b(x) = log_c(x) / log_c(b) for any valid c.
Read the common logarithm bases
Base 10 is the common logarithm, base e is the natural logarithm, and base 2 appears often in computing. Each base is allowed only when greater than 0 and not equal to 1. Arguments must be positive. Zero and negative arguments make the real logarithm undefined.
QuickCalculators rejects log of zero and log of a negative number with a field-level error. Bases such as 1 or negative bases are likewise rejected for the real-valued log used on this page.
Frequently asked questions
What is a logarithm?
A logarithm is the exponent that produces a given argument from a given base. In symbols, log_b(x) = y means b^y = x. The calculator solves for base, argument, or exponent under that definition.
How do you solve log base b of x equals y?
To solve log base b of x equals y, rewrite as b^y = x and evaluate the missing piece. If y is unknown, compute the log; if x is unknown, raise the base to y; if b is unknown, solve the exponential for the base.
What is the natural logarithm?
The natural logarithm is the logarithm with base e, written ln(x). It answers the question: e to what power equals x? Natural log mode on this page sets that base explicitly.
What does log mean without a base?
Without a base, log often means base 10 in school settings and base e in programming and many texts. The meaning is conventional, not universal. This calculator avoids the ambiguity by naming the base or using ln.
What are the log rules?
The log rules include the product rule, quotient rule, power rule, and change-of-base formula. They rewrite logs of combined expressions into sums, differences, multiples, or ratios of simpler logs. The page can show those rewrites when relevant.
Why is log of zero undefined?
Log of zero is undefined in the reals because no real exponent makes a positive base equal zero. Negative arguments are likewise outside the real log domain. The tool reports those inputs as errors rather than inventing a value.
Summary
The calculator solves log_b(x) = y for any missing part and treats every log answer as an exponent. Examples such as log_2(32) = 5 and log_10(1000) = 3 follow the definition directly. An unlabeled log is ambiguous between base 10 and base e, so the base must be stated.
Product, quotient, power, and change-of-base rules support simplification, while zero and negative arguments stay undefined.