Exponential Functions & Graphs PDF

Summary

This document provides a detailed explanation of exponential functions including growth, decay, asymptotes, and intercepts. Examples and exercises are included, specifically relating to logarithmic and exponential functions with clear definitions and formulas.

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# 4.1 Exponential Functions and Their Graphs The exponential function f with base a: f(x) = a<sup>x</sup> > * **Exponential Growth** a > 1 * **Exponential Decay** 0 < a < 1 **Asymptotes:** * **Horizontal Asymptote:** as x→∞ if f(x)→b then y=b is the horizontal asymptote. * **Vertical Asympt...

# 4.1 Exponential Functions and Their Graphs The exponential function f with base a: f(x) = a<sup>x</sup> > * **Exponential Growth** a > 1 * **Exponential Decay** 0 < a < 1 **Asymptotes:** * **Horizontal Asymptote:** as x→∞ if f(x)→b then y=b is the horizontal asymptote. * **Vertical Asymptote:** f(x)→∞, if xcthen x=c is a vertical asymptote *none for exponential* **Ex 1) Graph the exponential function g(x) = 2<sup>x</sup> by hand. Identify any asymptotes and intercepts. State the domain and range.** y=(½)<sup>x</sup> > * **Horizontal Asymptote**: y=0 * **Domain**: (-∞, ∞) * **Range**: (0, ∞) * **x-int**: *NONE* * **y-int**: 1 **Graphs of Exponential Functions:** The exponential function f(x) = a<sup>x</sup>, (a > 0, a ≠ 1) has * **Domain:** R * **Range:** (0, ∞) * **Horizontal Asymptote:** y=0 * **y-intercept:** (0, 1) * **the points:** (0, 1), (1, a), (2, a<sup>2</sup>), (-1, 1/a) **Ex 2) Use transformations to graph the exponential function f(x) = (3)<sup>x-1</sup> - 1 by hand. Identify any asymptotes and intercepts. State the domain and range.** * **Domain**: (-∞, ∞) * **Range:** (-1, ∞) * **x-int:** 0 * **y-int:** 0 # 4.2 Natural Exponential Function Any positive number can be used as the base for an exponential function, but some bases are used more frequently than others, ie. 2, 10, and e * **e can be approximated by (1 + 1/n)<sup>n</sup> for large values of n.** ## "Natural base": e ≈ 2.71828182845904523536.... | n | (1 + 1/n)<sup>n</sup> | |---|---| | 1 | 2.00000 | | 5 | 2.48832 | | 10 | 2.59374 | | 100 | 2.70481 | | 1000 | 2.71692 | | 10,000 | 2.71815 | | 100,000 | 2.71827 | | 1,000,000 | 2.71828 | ## Natural Exponential Function: The natural exponential function f(x) = e<sup>x</sup> with base e. It is often referred to as the exponential function. Since 2 <e <3, the graph of the natural exponential function lies between the graphs of y = 2<sup>x</sup> and y = 3<sup>x</sup> **Ex 3) Use transformations to graph the exponential function f(x) = -2(x-3)<sup>x</sup> +5 by hand. Identify any asymptotes and intercepts. State the domain and range.** * **Domain**: (-∞, ∞) * **Range**: (-∞, 5) * **x-int**: +5 * **y-int**: -2<sup>-3</sup>+5 **Ex 4) Evaluate each expression, round to 4 decimal places** * **i) 2<sup>3</sup> ≈ 8** * **ii) 3<sup>3</sup> ≈ 27** * **iii) e<sup>3</sup> ≈ 20.0855** * **iv) 2e<sup>-.53</sup> ≈ 1.1772** * **v) e<sup>4.8</sup> ≈ 121.5104** # Formulas for Compound Interest: ### COMPOUND INTEREST: Compound interest is calculated by the formula: A(t) = P(1 + r/n)<sup>nt</sup> where: * A(t) = amount after t years * P = principal * r = interest rate per year * n = number of times interest is compounded per year * t = number of years * **annually**: n = 1 * **semiannually**: n = 2 * **quarterly**: n = 4 * **monthly**: n = 12 * **daily**: n = 365 **Ex 8) If $4000 is borrowed at a rate of 16% interest per year, find the amount due at the end of 10 years if the money is compounded.** * **a) annually** A = 4000(1 + 0.16/1)<sup>10*1</sup> = $17646.74 * **b) monthly** A = 4000(1 + 0.16/12)<sup>10*12</sup> = $19603.76 * **c) daily** A = 4000(1 + 0.16/365)<sup>10*365</sup> = $19805.19 Let's see what happens as n increases indefinitely. If we let m = n/r, then: (1 + 1/m)<sup>m</sup> We know that as m becomes large, the quantity (1 + 1/m)<sup>m</sup> approaches the number e. Thus the amount approaches: **A = Pe<sup>rt</sup>** This expression gives the amount when the interest is compounded at "every instant." **For continuous compounding:** A = Pe<sup>rt</sup> **d) Find the amount after 10 years if $4000 is invested at an interest rate of 16% per year, compounded continuously.** A= 4000e<sup>0.16(10)</sup> = $19800.14 **Ex 9) Which of the given rates and periods would provide the best investment?** * **a) 2.5% compounded semiannually?** A = 4000(1 + 0.025/2)<sup>10*2</sup> = $2564 * **b) 2.25% compounded monthly?** A = 4000(1 + 0.0225/12)<sup>10*12</sup> = $2504 * **c) 2% compounded continuously?** A = 4000e<sup>(0.02)(10)</sup> = $2442 **2.5% compounded semiannually** # 4.3 Logarithmic Functions and Graphs For any exponential function f(x) = a<sup>x</sup>, its inverse is called a logarithmic function, base a: ## Definition of the Logarithmic Function: Let a be a positive number with a ≠1. The logarithmic function base a, denoted by log<sub>a</sub>, is defined by: > log<sub>a</sub> x = y ↔ a<sup>y</sup> = x **Ex 1) Solve for w.** * **a) log<sub>2</sub>√32 = w ↔ 2<sup>w</sup> = √32** * **b) log<sub>3</sub> 6 = 3 ↔ 3<sup>3</sup> = 6** * **c) log<sub>5</sub> w = 2 ↔ 5<sup>2</sup> = w** * **d) log<sub>√</sub>4 = w ↔ (√4)<sup>w</sup> = 4** **Solve for y.** * **(e) log<sub>2</sub> 1 = y ↔ 2<sup>y</sup> = 1** * **(f) log<sub>a</sub> d = y ↔ a<sup>y</sup> = d** **Convert to logarithmic form:** log<sub>a</sub> b = c ↔ a<sup>c</sup> = b * **(g) 2<sup>3</sup> = 8 ↔ log<sub>2</sub> 8 = 3** * **(h) e<sup>2</sup> = n ↔ log<sub>e</sub> n = 2 ↔ ln n = 2** ## Properties of Logarithms * **1. log<sub>a</sub> 1 = 0** We must raise a to the power of 0 to get 1. * **2. log<sub>a</sub> a = 1** We must raise a to the power of 1 to get a. * **3. log<sub>a</sub> x<sup>n</sup> = n log<sub>a</sub> x** We must raise a to the power of x to get a<sup>x</sup> * **4. a<sup>log<sub>a</sub> x</sup> = x** log<sub>a</sub> x is the power to which a must be raised to get x ## Common logarithm = log base 10 ↔ log<sub>10</sub> 3 = log 3 ≈ 0.4771 ## Natural logarithm = log base e ↔ log<sub>e</sub> 7 = ln7≈ 1.9459 **Ex 2. Evaluate each expression.** * **a) log<sub>5</sub> 5 = 1** * **b) 3<sup>log<sub>3</sub> 5</sup> = 5** * **c) log<sub>3</sub> 27 = 3** * **d) log<sub>5</sub> 1 = 0** **Use your calculator to approximate to 4 decimal places** * **(a) log 0.0000005 ≈ -6.301** * **(b) log 1,000,000 ≈ 6** * **(c) ln 17 ≈ 2.833 ** * **(d) ln(-3) = undefined** **Ex 1) Use the Laws of Logarithms to expand the expression completely:** log<sub>a</sub> (A/B) = log<sub>a</sub> A -log<sub>a</sub> B **Ex 2) Use the Laws of Logarithms to expand the expression completely:** log<sub>a</sub> (a<sup>2</sup> / (b<sup>3</sup>c<sup>5</sup>)) = 2log<sub>a</sub> a -3log<sub>a</sub> b -5log<sub>a</sub> c **Ex 3) Express as a single logarithm:** 5log<sub>a</sub> x -log<sub>a</sub> y + ½ log<sub>a</sub> z = log<sub>a</sub> (x<sup>5</sup> / (yz½)) **Ex 4) Express as a single logarithm:** log<sub>2</sub>x - 3log<sub>2</sub>y - 4 = log<sub>2</sub>(x / (y<sup>3</sup> * 16)) **Ex 5) Use your calculator to approximate to 4 decimal places** * **(a) log<sub>5</sub> 8 ≈ 1.292** * **(b) log<sub>7</sub> 2 ≈ 0.3155** **Ex 6) Find the exact value of log<sub>2</sub> 3.log<sub>3</sub> 8** log<sub>2</sub> 3.log<sub>3</sub> 8 = log<sub>2</sub> 3.log<sub>3</sub> 2<sup>3</sup> = 2.3 log<sub>2</sub> 3.log<sub>3</sub> 2 = 3 **Ex 7) Let log<sub>2</sub> = P and log<sub>3</sub> = Q, find** * **(a) log<sub>2</sub> 6 = P+Q** * **(b) log<sub>2</sub> 3 = Q-P** * **(c) log<sub>2</sub> 24a<sup>3</sup> = 3Q + 3log<sub>2</sub> a** * **(d) log<sub>3</sub> 2 ≈ 0.631**

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