Logarithmic equation

Definition

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.

5 Must Know Facts For Your Next Test

1. The basic form of a logarithmic equation is $\log_b(x) = y$, which can be rewritten in exponential form as $b^y = x$.
2. 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: $\log_b(M^k) = k \log_b(M)$.
3. Equations with different bases can be solved by using the change-of-base formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$.
4. Logarithmic equations must have arguments that are positive real numbers, meaning you should check for extraneous solutions.
5. 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.

Review Questions

• How do you convert a logarithmic equation to its equivalent exponential form?
• What is the change-of-base formula for logarithms?
• What are three key properties of logarithms used to simplify and solve logarithmic equations?

Related terms

Exponential Equation:

An equation in which variables appear as exponents. It can often be solved by taking the natural log or common log of both sides.

Natural Logarithm:

$\ln(x)$, a logarithm with base $e$, where $e$ is approximately equal to 2.71828.

Logarithm:

The inverse operation to exponentiation, indicating what power a base number must be raised to produce a given number.