1. Let's start with logarithmic functions. A logarithm answers the question: to what power must we raise a base number to get another number? The general form is $\log_b(x) = y$ which means $b^y = x$. 2. Important rules for logarithms include: - $\log_b(xy) = \log_b(x) + \log_b(y)$ - $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ - $\log_b(x^r) = r \log_b(x)$ 3. Now, trigonometric functions relate angles of a triangle to ratios of sides. The main functions are sine ($\sin$), cosine ($\cos$), and tangent ($\tan$). 4. For an angle $\theta$ in a right triangle: - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ 5. These functions are periodic and have important properties such as: - $\sin^2(\theta) + \cos^2(\theta) = 1$ - $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ 6. Both logarithmic and trigonometric functions are fundamental in many areas of math, physics, and engineering.