$\oint 7$ The Dreaded Logs

$\oint 7.1$ & $7.2$ Exponential Growth and Decay

$$ f(x) = a \cdot b^x $$

$0 < b < 1 \to$ Decay

$b > 1 \to$ Growth

Example:

$$ A(t) = P(1+\frac{r}{n})^{nt} $$

$a = P$

$b = 1 + \frac{r}{b}$

$x = nt$

$(1 + \frac{r}{n}) > 1 \to$ Growth

Exponential Growth/Decay Formulas

Growth: $$ A(t) = A_0(1+r)^t$$

Decay: $$ A(t) = A_0(1-r)^t$$

Example: Snowmobile cost 4200 $. Value decreases by 10% per year. What is the value after 3 years?

$$ A(3) = 4200(1-.1)^3$$

Transformation:

$$ f(x) = a(b)^{x - h} + k $$

$\oint 7.3$ Functions involving $e$

$$ f(x) = e^x $$$$\lim_{n\to\infty}\bigg(1+\frac{1}{n}\bigg)^n = e$$

Tiger Shark length (cm) is modeled by:

$$ l = 337 - 276e^{-0.178t}$$

$\oint 7.4$ Logarithms...

$$ b_x = y \Leftrightarrow log_by=x $$

$\oint 7.5$ Properties of Logarithms...

Log of a product

$log_bm + log_bn = log_b(mn)$

Log of a quotient

$log_bm - log_bn = log_b(\frac{m}{n})$

Log of a power

$log_bm^n = n \cdot log_bm$

Change of base

$log_mn = \frac{log_bn}{log_bm}$

$\oint 7.6$ Solving Exponential and Logarithmic Equations