A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. Solving these equations often requires using properties of logarithms or converting them to exponential form.
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The basic form of a logarithmic equation is $\log_b(x) = y$, which can be rewritten in exponential form as $b^y = x$.
To solve logarithmic equations, you often need to use properties such as the product rule: $\log_b(MN) = \log_b(M) + \log_b(N)$, the quotient rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$, and the power rule: $\log_b(M^k) = k \log_b(M)$.
Equations with different bases can be solved by using the change-of-base formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$.
Logarithmic equations must have arguments that are positive real numbers, meaning you should check for extraneous solutions.
Common methods to solve logarithmic equations include isolating the logarithm on one side and then exponentiating both sides or combining multiple logs into a single log.