Achal, Andrew G., Chao, Michelle, William June 6, 2014
Chapter 8 Study Guide
Honors Math 3 group assignment Making a unit study guide Logarithmic and Exponential Functions This study guide serves to help you study for the Final Exam by aggregating the most important lessons from Chapter 8, on logarithmic and exponential functions. Here is a list of the most important concepts to jog your memory: ● Definition of logarithmic and exponential functions. (MICHELLE) ○ Shape / graph of exponential and logarithmic functions, including transformations. ○ The inverse of an exponential function is a logarithmic function, and each function has its own graphical properties. ● Interest calculations and formulas, e, and limits for certain functions. (ACHAL) ○ Calculating periodically compounded interest. ○ Calculating continuously compounded interest. ● Exponent properties, logarithmic properties and their proofs, and their applications. (ALL) ○ The “change of base” rules for logs and exponents and how to prove them. ○ The “Fundamental Law of Logarithms” and their corollaries. ● Finding recursive and closed-­form function formulas for exponential equations. (ANDREW) ○ Using log(x) to find the number of years for growth to occur. ● Representing logarithmic, power, and exponential functions linearly on log paper. (WILLIAM) ○ Logarithmic, power, and exponential functions can be represented linearly using special graphs with logarithmic scales and can be derived from tables by rewriting x and y columns in terms of logs. ○ Interpretation and usage of an exponential ruler. ○ Deriving equations from tables. ○ Proving why certain functions look linear on certain types of log paper. Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide Key List of Formulas, Definitions, and Theorems ● Exponential Function -­-­ f(x) = a*bx where a does not equal 0 and b is positive and does not equal 1 ● Logarithmic Function -­-­ the inverse of an exponential function, written f(x) = logb(x) ● Theorem 8.5-­ When the x axis is linear and the y axis is logarithmic, the graph of an exponential function y = a*bx with a > 0 and b > 0, appears as a linear line. (Proof on p. 652) ● Logarithmic Function as a Straight Line Theorem -­-­ When the x axis is logarithmic and the y axis is linear, the function y = logb(x) appears as a linear line. (From homework on p.654 part a) ● Power Function as a Straight Line Theorem -­-­ When both the x and y axes are logarithmic, the function y = a*xb appears as a linear line. (From homework on p.654 part b) ● Fundamental Law of Exponents -­-­ f(x+y) = f(x)*f(y) let f(x) = bx ○ Corollaries: f(x-­y) = f(x)/f(y) ( or bx-­y = bx/by ) f(xy) = (f(x))y ( or bxy = (bx)y ) ● Other Properties of Exponents: ○ let f(x) = bx f(0) = 1 f(-­x) = 1/f(x) ● Change of Base Rule for Logs (Theorem 8.4) -­-­ logx(y)= logb(y) / logb(x) ● Change of Base Rule for Exponents -­-­ cx = aloga(c)*x ● Fundamental Laws of Logarithms ■ logb(MN) = logb(M) + logb(N) ■ logb(M/N) = logb(M) -­ logb(N) ■ logb(Mp) = p logb(M) ● Other Properties of Logarithms logb(1) = 0 and logb(b) = 1 logb(1/N) = -­logb(N) logb(M) = logb(N)⇔M = N logb(bx) = x = blogb(x) Paragraph: In the expression b^n, b is called the base and n is called the exponent. For any number b and positive integer n, if b ≠ 0, then b^0 = 1;; b^1 = b;; if n>1 → b^n = b*b*b*b......b where b is a factor n times. b^(-­n) = 1/(b^n). In the laws of exponents, (b^m)(b^n) = b^(m+n);; (b^m)/(b^n) = b^(m-­n);; (b^m)^n = b^mn;; (b^m)(c^m) = (bc)^m. A logarithm is an exponent;; it is the inverse of an exponent. Instead of saying “2 to the 4th power equals 16,” logs would be more like “4 is the exponent to which the base 2 must be raised to equal 16.” If b is a positive number not equal to 1 and x>0, (log base b to the x) = y if and only if b^y = x. The change of base formula states that (log base b to the x) = (log base 10 to the x)/(log base 10 to the b). Laws of logarithms include, for any positive base b ≠ 1 and any positive numbers x, y, and n: 1| (log base b to the (xy)) = (log base b to the x) + (log base b to the y);; 2 | (log base b to the (x/y)) = (log base b to the x) -­ (log base b to Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide the y);; 3 | (log base b to the x to the n) = n*(log base b to the x);; 4 | (log base b to the b to the n) = n. Also in Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide Part 1: Logarithmic and Exponential Properties In the expression b^n, b is called the base and n is called the exponent. For any number b and positive integer n, if b ≠ 0, then b^0 = 1;; b^1 = b;; if n>1 → b^n = b*b*b*b......b where b is a factor n times. b^(-­n) = 1/(b^n). In the laws of exponents, (b^m)(b^n) = b^(m+n);; (b^m)/(b^n) = b^(m-­n);; (b^m)^n = b^mn;; (b^m)(c^m) = (bc)^m. A logarithm is an exponent;; it is the inverse of an exponent. Instead of saying “2 to the 4th power equals 16,” logs would be more like “4 is the exponent to which the base 2 must be raised to equal 16.” If b is a positive number not equal to 1 and x>0, (log base b to the x) = y if and only if b^y = x. The change of base formula states that (log base b to the x) = (log base 10 to the x)/(log base 10 to the b). Laws of logarithms include, for any positive base b ≠ 1 and any positive numbers x, y, and n: 1| (log base b to the (xy)) = (log base b to the x) + (log base b to the y);; 2 | (log base b to the (x/y)) = (log base b to the x) -­ (log base b to the y);; 3 | (log base b to the x to the n) = n*(log base b to the x);; 4 | (log base b to the b to the n) = n. Also in particular, (log base b to the 1) = (log base b to the b to the 0) = 0 and (log base b to the b) = (log base b to the b to the 1) = 1. For real life applications, a solution’s pH is defined by the logarithmic equation , where t is the hydronium ion concentration in moles per liter. Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide Part 2: Logarithmic and Exponential Function Definitions The inverse of an exponential function is a logarithmic function, and each function has its own graphical properties. An exponential function is a function in the form: f(x) = a*bx where a does not equal 0, b does not equal 1 and b is greater than 0. Exponential functions are one-­to-­one. The graph of an exponential function changes based on the values of a and b. If b > 1, the graph starts from its limit and increases or decreases from there. If 0 < b < 1, the graph either increases or decreases toward the limit. If a > 0, the graph is always above its limit. If a < 0, the graph is always below its limit. Because exponential functions are one-­to-­one, they have inverse functions. this inverse is represented by f(x)=logb(x). In this function, f(x) is the exponent such that b to the power of that exponent equals x. If logb(x) = y, then by=x. the expression log(x), without a base b, it is assumed that the base is 10. When written as ln(x), the base is e. Using the fact that logb(x)= log(x)/log(b) (change-­of-­base rule, described later), we can show all logarithmic functions as vertical stretches of each other. For logarithmic functions with a base greater than 1, the smaller the base, the more stretched the graph. For logarithmic functions with a base 1/b , the graph is the logarithmic graph of base b flipped over the x axis. In a logarithmic function, when the x is modified, horizontal transformations occur. For example, f(x)=log2(x/5) is a horizontal stretch (by 5) of the function log2(x). When y is modified (modifications to the end result), vertical transformations occur. For example, f(x) = log 2 (x)
3
is a vertical compression (by 3) of the function log2(x). Achal, Andrew G., Chao, Michelle, William June 6, 2014
Part 3: Interest & Mortgage
Honors Math 3 group assignment Making a unit study guide Interest Calculations Interest is a fee paid by an agent who borrows assets from an owner, with compensation for the use of the assets. When money is borrowed, interest is typically paid to the lender as a percentage of the principal amount (the original amount given to the borrower). The percentage of the principal that is paid as a fee over a certain period of time (monthly, yearly, etc.) is called the interest rate. We are going to focus on compound interest. Compound interest arises when interest is added to the principal amount, so that the interest that has been added also earns interest. The frequency of compounding is thus very important. There are three frequencies of compounding relevant to this lesson: periodically compounded interest, such as: monthly (the interest is compounded on a month-­to-­month basis), and annual (the interest is compounded on a year-­to-­year basis). The other type of compounding is continuously compounded interest (interest that is, hypothetically, computed and added to the balance of an account every instant). Periodically Compounded Interest: r
P * (1 + n ) nt P = initial amount invested (principal) r = APR expressed as a decimal (r = 0.06 if it’s 6%) n = number of compounding periods per year t = number of years of investment Continuously Compounded Interest: Pert P = initial amount invested (principal) e = euler’s number r = APR expressed as a decimal (r = 0.06 if it’s 6%) t = number of years of investment But what is this special e number? This is Euler’s number, which is defined as: | the limit of (1 + 1/n) nt as n gets larger and larger. Proof on proofs page. Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide Part 4: Finding recursive and closed-­form function formulas for exponential equations. Recursive and Closed-­form functions can be found for exponential functions. Recursive functions take the form: a if n = 0 B(n) = b * B(n-­1) if n > 0 a is the y-­intercept, so it is when n equals zero on the table or graph. To find b, if there are two inputs that differ by 1, you can find the ratio between the terms (y2/y1). Closed Form functions take the form: f(x) = a * bx There are two methods to find the closed form function. The first method is to find the ratio between two terms whose inputs differ by 1, and substituting that in for b. Then, plug in a given point in to the formula, and the equation is solved for a. The other method to finding the closed form function is to choose two points, and set up the equation f(x) = a* bx for each. Then divide the two equations, in order to build an equation for b. Solve for b. Then plug that into the equation, and find a. Examples of this can be found on page 603. Using log(x) to find the number of years for growth to occur: log is the inverse of the exponential equation. So, if you want to find how long it takes (x years) to get to number y, just find the log of the exponential function and put the number y as the input of the logarithmic function, and that will get the amount of years. Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide Part 5: Representing logarithmic, power, and exponential functions linearly on log paper Logarithmic, power, and exponential functions can be represented linearly using special graphs with logarithmic scales and consequently, can be derived in a problem if you’re only given a table by rewriting x and y columns in terms of logs. A logarithmic scale is a scale in which the denominations on the scale are differences in exponents for a certain number as shown below: The logarithmic scale is useful when trying to represent data with vastly different values and when used in a graph, depending on which axes uses the logarithmic scale, can represent a power, exponential, or logarithmic function linearly. This makes it useful to derive a logarithmic, power, or exponential function if only given a table depending on the relationship between the input output columns. If the input or “x” column of the table and output or “y” column share a linear relationship, then the function is linear. If the function is linear when the “y” column is re-­expressed in terms of logs, then that table is a table for an exponential function. When the function is linear when the “x” column is re-­expressed in terms of logs and the “y” column is as it is, then that table is a table for a logarithmic function. If the function is linear when both the “x” and “y” column are re-­expressed in logs, then that table is a table for a power function. Visit pages 648-­652 for more information. Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide Part 6: Review Questions 1.
A. Without using your calculator, fill in the missing values of the table of values for the log3(x) function. x log5(x) 1 2 0.4307 3 4 5 6 7 1.2091 8 9 1.3652 10 B. Using the values in the table, find the value of log5(80). C. Find the value of log3(6). (Please note that the base is 3, not 6). Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide 2. x log9(y) 1 9 3 10 9 11 27 12 81 13 243 14 A. Find an equation for this table that expresses y in terms of x. Use the scale on the bottom to do the following questions B. Use a logarithmic function to express the relationship between x and y. C. Use an exponential function to express the relationship between x and y. Achal, Andrew G., Chao, Michelle, William June 6, 2014
3. Honors Math 3 group assignment Making a unit study guide For all parts of the question, let f(x) = 7*2^(3x) A. Write a function for the inverse of f(x). 4.
5. B. Identify the geometric transformations that transform the graph of (log base 2 to the x) into the graph of the inverse of f(x). Using the following information, make both a recursive and closed-­form exponential function. q(1) = 19.2 q(3) = 196.608 A. Suppose that an initial investment of $600 grows with a 7% APR. If the interested is compounded daily, how much will the investment be worth after 1 year? B. Suppose that an initial investment of $7,000 grows with a 14% APR. If the interested is compounded monthly, how much will the investment be worth after 4 years? C. Suppose that it is compounded continuously -­ how much would the investment be worth after 3 years? Achal, Andrew G., Chao, Michelle, William June 6, 2014
Answer Key 1A. x log5(x) 1 0 2 0.4307 3 0.6825 4 0.8613 5 1 6 1.1132 7 1.2091 8 1.292 9 1.3652 10 1.4307 1B. log5(80) = log5(8*10) = log5(8) + log5(10) = 2.7227 1C. Use the change of base rule. log (6)
log3(6) = log 5(2) = 1.631 5
2A. y = x2/9-­9 2B. log17 (y)= 0.5x 2C. y = 170.5x 3A. f^-­1(x) = [(log base 2 to the x) -­ (log base 2 to the 7)]/3 3B. Horizontal stretch by 7, vertical compress by 3 4. Closed: 19.2 = a * b1 196.608 = a * b3 Divide. (196.608/19.2) = (a * b3/a * b1) 10.24 = b2 Honors Math 3 group assignment Making a unit study guide Achal, Andrew G., Chao, Michelle, William June 6, 2014
Honors Math 3 group assignment Making a unit study guide 3.2 = b Substitute. 19.2 = a * 3.21 6 = a Final Answer: f(x) = 6 * 3.2x Recursive: Using the above information, we know that the y-­intercept is 6 and the rate is 3.2, so... B(n) = 6 if n = 0 3.2*B(n-­1) if n > 0 5A. 600 * (1 + .06 / 365) (365*1) = $637.10 5B. 7,000 * (1 + .14 / 12) (12*4) = $12,205.15 5C. 7,000 * e (12*3) = $10,653.73