Logarithmic equation

Definition

A logarithmic equation is an equation that involves a logarithm with a variable inside its argument. Solving these equations typically requires using properties of logarithms and exponentials.

5 Must Know Facts For Your Next Test

1. The basic form of a logarithmic equation is $\log_b(x) = y$, where $b$ is the base and $x$ is the argument.
2. To solve a logarithmic equation, you often convert it to its exponential form: $\log_b(x) = y \Rightarrow b^y = x$.
3. Logarithmic properties such as the product rule, quotient rule, and power rule are essential for simplifying and solving logarithmic equations.
4. Extraneous solutions may arise when solving logarithmic equations, so it's crucial to check all potential solutions in the original equation.
5. Equations involving multiple logarithms can sometimes be simplified by combining them using properties like $\log_b(MN) = \log_b(M) + \log_b(N)$.

Review Questions

• What is the exponential form of the logarithmic equation $\log_2(8) = x$?
• How would you solve the logarithmic equation $\log_3(x - 1) = 2$?
• What property of logarithms allows you to combine $\log_b(A) + \log_b(B)$ into a single log expression?

Related terms

Exponential Equation: An equation in which variables occur as exponents. For example, $2^x = 8$.

Logarithm: The inverse function of exponentiation, denoted as $\log_b(x)$ for base $b$. It answers the question 'to what exponent must $b$ be raised to produce $x$?'

Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ is any positive number different from $1$. This formula allows conversion between different bases.