Logarithmic Equations
1. Let's start by stating what logarithmic equations are: equations that involve the logarithm of a variable.
2. A common form is $\log_a(x) = b$, which means $x = a^b$.
3. To solve logarithmic equations, we often rewrite them in exponential form to isolate the variable.
4. Example: Solve $\log_2(x) = 3$.
5. Rewrite in exponential form: $x = 2^3$.
6. Calculate $2^3 = 8$.
7. So, the solution is $x=8$.
8. Another example: Solve $\log(x+1) = 2$ where the base is 10 (common log).
9. Rewrite: $x+1 = 10^2$.
10. Calculate $10^2 = 100$.
11. Solve for $x$: $x = 100 - 1 = 99$.
12. Remember to check the domain: arguments of logarithms must be positive.
13. For example, $x+1 > 0$ so $x > -1$; $x=99$ satisfies that.
14. In summary, solve by rewriting the log equation in exponential form, then solve and check domains.
15. If you have a specific logarithmic equation, feel free to share it for a detailed solution.