Using the Logarithmic Equation Calculator
Use this for one logarithm written as log_base(ax + b) = c. Enter a positive base different from one, integer coefficients a and b, and a numerical target c. Convert the logarithm to exponential form before isolating x. This tool does not combine multiple logarithms or solve logarithmic inequalities.
Worked example
Use the example inputs below to reproduce this result. These are the form's starting values, not recommended targets. Changing your inputs updates your result above; this worked example stays fixed for comparison. Results are rounded for display.
Example inputs
- Logarithm base (positive, not 1)
- 10
- a in argument ax + b
- 2
- b in argument ax + b
- 1
- c (target)
- 2
Example result
- Solution
- x ≈ 49.5
- x
- 49.5
- Positive argument check
- 100
- Verification
- log base 10 of the substituted argument ≈ 2
- Steps
- Start with log base 10 of (2x + (1)) = 2. Require 2x + (1) > 0. Convert to exponential form: 2x + (1) = 10^(2) ≈ 100. Subtract 1 and divide by 2: x ≈ 49.5. Substitute back and verify that the argument is positive.
Method and assumptions
Convert to ax+b=base^c and solve for x, then check the original logarithm domain. Integer a,b from −10,000 to 10,000; decimal base and target are allowed. Results are approximate. Overflow, underflow and unstable cancellation are rejected rather than returning an unreliable root. One logarithm only.
Reference: symbolab.com — method and further reading
Check the answer in the original problem
After solving, substitute the result into the original equation or relationship where possible. This can catch a sign error or a coefficient entered in the wrong position. Keep the exact inputs available when comparing with another tool, since changing a coefficient or rounding an intermediate value can change the problem being solved.
Common question
Why is a base of one invalid?
Every power of one equals one, so it cannot define the inverse function required for a logarithm. Real logarithms require a positive base different from one and a positive argument.