Logarithmic Equations Calculator
Solve log(x−3) + log x = 1 and the algebra hands you two roots: 5 and −2. Only one of them is an answer.
Put −2 back and the first term asks for log(−5), which does not exist. That root is extraneous, and losing marks to it is what makes these questions harder than they look. This logarithmic equations calculator checks every root against the original equation and shows you which one fails and why.
How to Solve Logarithmic Equations
Every question of this kind reduces to one of two shapes, and getting to one of them is most of the work.
One log equals a number. log(A) = c. Raise the base to both sides and the logarithm disappears: A = 10ᶜ for log, or eᶜ for ln. log(x+1) = 2 becomes x + 1 = 100, so x = 99.
One log equals another log. log(A) = log(B). The same logarithm of two things can only be equal when the two things are equal, so A = B and the logs simply drop away.
If there is more than one log on a side, condense first — adding logs multiplies the insides, subtracting divides them, and a coefficient goes up as a power. Only then undo it.
Why Extraneous Solutions Appear
They are not a quirk. They are built into the method, and knowing why makes them easy to catch.
log(x−3) + log x = 1 only makes sense when x > 3, because both logs need a positive inside. Condense it to log(x²−3x) = 1 and that restriction quietly vanishes — x²−3x is positive for x < 0 as well. The condensed equation allows values the original never did, and −2 is one of them.
So the algebra is not wrong; it is answering a slightly wider question. Putting each root back into the equation you started with is what narrows it again. Solving log equations without that step is only half the method.
How to Use This Logarithmic Equations Calculator
Type the equation and press =. The keypad has log, ln, x and the equals sign, so log(x−3) + log x = 1 is a handful of taps. Both log and ln work, as long as every logarithm in the equation is in the same base.
The answer appears with a note saying how many roots were thrown out. One tap opens every root that came out of the algebra — the ones that survived in green, the ones that failed struck through, each with the exact log that broke and the number it was asked for.
Answers stay exact where they can: log x + log 3 = 1 gives 10/3, not 3.33. Under the working you will find the same five steps you would write on paper, including the check. If you need to join the logs by hand first, the condense logarithms calculator does that part, and the log calculator handles plain values in any base. It is free to use online, with no account.
Logarithmic Equations FAQ
How do you solve a logarithmic equation?
What is an extraneous solution in a logarithmic equation?
Why do you have to check the answers?
Can a logarithmic equation have no solution?
How do you solve log x = 2?
Condense, undo, solve, and then check — and it is the checking that separates a right answer from a plausible one.