Logarithmic Equations
Grade 10 Algebra — A Complete Guide
1. Definition
A logarithmic equation is an equation that contains a logarithm of an expression involving the unknown variable.
Solving a logarithmic equation means finding all values of $x$ that:
- ✅ Satisfy the equation algebraically
- ✅ Belong to the domain of the equation
🔑 Domain Condition: All arguments of logarithms must be strictly positive. If $\log_a(\text{expression})$ appears, then $\text{expression} > 0$. Also, the base $a > 0$ and $a \neq 1$.
📖 Key Vocabulary
Logarithmic Equation Equation with log of unknown
Domain Condition Arguments must be > 0
Argument Expression inside the log
Extraneous Root Solution that fails domain check
Change of Base Converting log to different base
Substitution Replacing log expression with variable
Product Rule $\log_a(MN) = \log_a M + \log_a N$
Quotient Rule $\log_a\!\left(\frac{M}{N}\right) = \log_a M - \log_a N$
Power Rule $\log_a(M^k) = k\,\log_a M$