1. Let's start by stating the problem: understanding logarithmic and exponential equations. 2. Exponential equations have the form $y = a^x$, where $a$ is a positive constant not equal to 1, and $x$ is the exponent. 3. Logarithmic equations are the inverse of exponential equations and have the form $y = \log_a(x)$, which means $a^y = x$. 4. Important rules: - $a^{m+n} = a^m \times a^n$ - $a^{m-n} = \frac{a^m}{a^n}$ - $\log_a(xy) = \log_a(x) + \log_a(y)$ - $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$ - $\log_a(x^k) = k \log_a(x)$ 5. To solve exponential equations, you often take the logarithm of both sides to bring down the exponent. 6. To solve logarithmic equations, you rewrite them in exponential form to isolate the variable. 7. Example: Solve $2^x = 8$ - Since $8 = 2^3$, we have $2^x = 2^3$ - Therefore, $x = 3$ 8. Example: Solve $\log_2(x) = 3$ - Rewrite as $2^3 = x$ - So, $x = 8$ Understanding these concepts helps in solving many real-world problems involving growth, decay, and scales.