Lecture 5: Exponential Functions

f(x)=axf(x) = a^x a>0,a1\rightarrow a > 0, a \neq 1

D=R,R=(0,)D = \mathbb{R}, R = (0, \infty)

*aa cannot be negative because if xx is a fraction, you can't root a negative #

^if x=0,a0=1x = 0, a^0 = 1

if x=n,an=1anx = -n, {a^{-n}} = \frac{1}{a^n}

if x=1n,a1n=anx = \frac{1}{n}, {a^{\frac{1}{n}}} = \sqrt[n]{a}

if x=pq=apq=apqx = \frac{p}{q} = {a^{\frac{p}{q}}} = \sqrt[q]{a^p}

  1. xaxb=xa+bx^a x^b = x^{a + b}
  2. xaxb=xab\frac{x^a}{x^b} = x^{a - b}
  3. (xa)b=xab(x^a)^b = x^{ab}
  4. axbx=(ab)xa^x \cdot b^x = (a \cdot b)^x or axbx=(ab)x\frac{a^x}{b^x} = (\frac{a}{b})^x

*if a>0a > 0 and b>0b > 0 applies to all real numbers xx & yy

Exponential Function \rightarrow f(x)=10xf(x) = 10^x

Power Function g(x)=x10\rightarrow g(x) = x^{10}

Interest Example: In the year 2000, you invest $100 into a savings account w/ 5.5% interest compound annually. Find the formula for amount of $ in xx years.

Natural Exponential Functions exe^x

f(x)=exf(x) = e^x where e=2.718e = 2.718... (an irrational #)

D=RD = \mathbb{R} & R=(0,)R = (0, \infty)

Exponential Growth & Decay

f(x)=Q0eKxf(x) = Q_0 e^{Kx} where Q0Q_0 is a constant and if K>0K > 0 = growth, K<0K < 0  = decay


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