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Logarithmic Equation Solver

Enter the base and right-hand value to solve log_b(x) = y.

How it is calculated

The equation logb(x) = y is solved by rewriting it in exponential form: x = by. You supply the base b and the right-hand value y, and the tool raises b to the power y directly, with no iteration. The base must be positive and not equal to 1; a base of 1, zero or a negative number gives no result. The exponent y may be negative or fractional. The answer is shown with six decimals, but if it is 1,000,000,000 or larger, or smaller than 0.0001, it switches to scientific notation with six significant digits (for example 1.0 × 10¹²). The verification line only restates the base, the displayed x and y in the form logb(x) = y; it does not recompute the logarithm.

Worked example

Solve log2(x) = 10. Here b = 2 and y = 10, so x = 210. Doubling ten times gives 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. The tool displays 1,024.000000 and writes the check line as log_2(1,024.000000) = 10.0000. In words: 2 must be raised to the power of 10 to give 1,024. The same method with base 10 and y = −2.5 would give 0.003162.

Try this example →

Common mistakes

  • Entering the argument x instead of the result. This tool is given the base and the value y of the logarithm, and it returns x; to find y from a known x you need a logarithm calculator instead.
  • Using base 1 or a negative base. Neither is a valid logarithm base, so the tool returns nothing.
  • Reading the verification line as an independent check. It restates the numbers the tool just produced, so it will look correct even if you typed the wrong y.
  • Dropping the minus sign on y. A negative y gives a value between 0 and 1: base 10 with y = −2.5 gives 0.003162, not 316.2.
  • Treating a scientific-notation result such as 1.0 × 10¹² as just 1.0, or ignoring the exponent. It means 1 × 1012, a 1 followed by twelve zeros.

About this calculator

A logarithmic equation of the form log_b(x) = y asks: to what power must the base b be raised to produce x? By the definition of a logarithm as the inverse of exponentiation, the equation rearranges directly into x = b^y, which is exactly how this calculator solves it — no iteration or approximation needed, just applying the inverse relationship.

This calculator also writes the result back as log_b(x) = y with your base, x and y filled in, so you can check the equation yourself (it restates the values rather than recomputing the logarithm), which is good practice since logarithms are only defined for positive x and a positive base b ≠ 1. Solving log equations this way underlies real-world calculations such as pH from hydrogen-ion concentration, decibel and Richter-scale conversions, and finding an unknown in exponential growth or decay models like compound interest or radioactive half-life.

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Comments (1)

  • Editorial team

    One handy use is converting a pH reading back to a concentration. With base 10 and y set to -7.4, the tool returns 3.98107 × 10⁻⁸, the hydrogen-ion concentration in mol/L for a pH of 7.4. The same move works for decibel ratios, where you scale by ten or twenty first. Which scale do you usually need to convert back from: pH, decibels or Richter?

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