Free Logarithm Equation Calculator

This logarithm equation calculator solves logb(x) = y for whichever variable you leave out. Enter any two of the base, the argument, or the exponent, and it works out the third, with the full step-by-step solution shown underneath.

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Logarithm Equation Solver

logb(x) = y

Enter any 2 values to solve for the third

Rules for Each Field

  • b (base) must be positive, and can't equal 1.
  • x (argument) must be positive — logarithms of zero or negative numbers aren't defined.
  • y (exponent) can be positive, negative, zero, or a decimal.

Answer

Step-by-Step Solution:

Share This Calculation

Opening this link fills in your values and shows this result automatically.

Enter any 2 of b, x and y and the third, worked out step by step, will appear here.

How to Use the Logarithm Equation Calculator

  1. Fill in any two of the three fields — b (base), x (argument), or y (exponent) — and leave the one you want to solve for blank.
  2. Click Calculate. The missing value appears at the top, with the full working shown underneath so you can see exactly how it was solved.
  3. Check the rules panel if you get an error — the base can't be 1, and the base and argument both need to be positive.
  4. Use Copy, Print or Share to grab the result — Copy puts the plain-text answer on your clipboard, Print gives you a clean printout of just your equation and the steps, and Share generates a link that reloads with your exact values and result already filled in.
  5. Click Clear to reset all three fields and start over.

Practical Example / Formula Used

The equation logb(x) = y is just another way of asking "b to what power gives x?" — and y is that power. Depending on which value is missing, this calculator rearranges the equation one of three ways:

Solve for y: y = log(x) ÷ log(b)
Solve for x: x = by
Solve for b: b = x(1/y)

Worked example — solve log8(4) = y:

y = log(4) ÷ log(8)
y = 0.60206 ÷ 0.90309
y = 0.66667

You can check this: 8 raised to the power of 0.66667 comes out to almost exactly 4, confirming the answer.

Why Use a Logarithm Equation Calculator? / Benefits

  • Saves you from switching bases by hand. Most calculators only have log base 10 or base e built in, so solving log8(4) directly isn't possible without the change-of-base formula this tool applies automatically.
  • Works in all three directions. You're not stuck solving for the exponent — it just as easily solves for the base or the argument if that's the unknown in your problem.
  • Shows the reasoning, not just the number. Seeing every step makes it useful for checking homework or actually understanding where an answer came from, not just copying a result.
  • Catches invalid equations early. If your numbers would require the log of a negative number or a base of 1, you get a clear explanation instead of a confusing wrong answer.

Key Factors / Variables Explained

  • b (base): The number being raised to a power. Must be positive and can't be 1, since 1 raised to any power always equals 1, which breaks the equation.
  • x (argument): The result of raising the base to the exponent. Must be positive, since no real power of a positive base ever produces zero or a negative number.
  • y (exponent): The power the base is raised to. This is the only one of the three that can be zero, negative, or a decimal without any restriction.

A Second Worked Example: Solving for the Base

If logb(16) = 2, find b:

b = 16(1/2)
b = √16
b = 4

Checking: log base 4 of 16 does indeed equal 2, since 4² = 16.

Frequently Asked Questions (FAQs)

1 raised to any power always equals 1, no matter what the exponent is. That means log base 1 of any number other than 1 has no solution, and log base 1 of 1 has infinitely many, so a base of 1 simply isn't allowed in a logarithm.

No real power of a positive base ever lands on zero or a negative number, so there's no real value of y that could make b to the power of y equal a negative x. That's why the argument of a logarithm always has to be positive in the real numbers.

It's logb(x) = log(x) ÷ log(b), using logs of any common base (this calculator uses base 10) on the top and bottom. It's what lets you compute a logarithm in any base using only the log button on a standard calculator.

Yes, whenever x is smaller than 1 while the base is greater than 1 (or vice versa). For example, log base 2 of 0.5 equals negative 1, since 2 to the power of negative 1 is one half.

No. The link only encodes the exact values you entered, so anyone opening it sees the same calculation you did, nothing more.

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