Logarithmic Equations
Solve the equation over ℝ. If it has only one solution, leave the second box empty. If it has no solution, tick the box.
Correct answers
To bronze: 0 of 5
Daily streak
To Busy Bee: 0 of 3 days
Today: 0 of 5 exercises
In a logarithmic equation the unknown is in the argument of a logarithm, for example log₂ (x + 7) = 2. By the definition, x + 7 = 2² = 4, so x = −3. Careful: a logarithm exists only for a positive number. That is why we write down the condition at the start and check every root at the end.
What does this generator do?
On the easy level you solve equations like log₂ (x + 7) = 2 directly from the definition. On the medium level the equation contains a sum or difference of logarithms — one root often fails the condition, and sometimes the equation has no solution at all. On the hard level you solve equations using the substitution t = log x and equations like log (x² + e) = log (fx + g).
How to use it
1) Write down the condition: every argument of a logarithm must be positive. 2) Use the laws of logarithms to combine them into one. 3) Remove the logarithm: log_a A = c means A = a^c, and log_a A = log_a B means A = B. 4) Solve the resulting equation. 5) Check every root against the condition — discard those that fail.