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Solve logarithmic equations with step-by-step solutions and domain checking
If logb(x) = r, then x = br
Use when you have a single log equal to a number
Condense multiple logs using product/quotient rules first
Use for equations with log(x) + log(y) or log(x) - log(y)
If logb(A) = logb(B), then A = B
Use when logs with same base are on both sides
Log arguments must be positive: x > 0
Reject any solution where the argument is zero or negative
For logb(x) = r, convert to exponential form: x = br. Then check that x > 0. For example, log(x) = 2 means x = 10² = 100.
If the logs have the same base, you can set the arguments equal. For example, if log(2x) = log(x + 5), then 2x = x + 5, so x = 5. Always verify the domain afterwards.
First condense using log properties. For log(x) + log(y) = r, condense to log(xy) = r, then solve xy = br. Remember to check that all arguments remain positive.
You can't take the logarithm of zero or negative numbers (in real numbers). When solving, you might get algebraically valid solutions that violate the domain. These "extraneous solutions" must be rejected.
Some logarithmic equations have no real solution. This happens when the algebraic solution violates the domain (negative or zero argument) or when the equation structure is impossible.
Yes, but first convert all logs to the same base using the change of base formula. Then proceed with the standard methods. Usually it's easiest to convert everything to natural logs.
After condensing and converting to exponential form, you might get a quadratic equation. Solve using factoring, completing the square, or the quadratic formula. Check both solutions in the original equation.
Common errors: Forgetting domain restrictions, incorrectly applying log properties (especially log(x+y) ≠ log(x) + log(y)), and not checking solutions in the original equation.