E XPONENTS AND L OGARITHMS AP Calculus AB Summer Review

E XPONENTS An exponent is the operation of multiplying the same base number several times The repetition is represented by the exponent: Ex: 2 AP Calculus AB, SJHS

S IGNS OF E XPONENTS Since the repetition signifies the times you multiply the base number, this must be a positive number We know from basic rules of division, however, that numbers in the denominator reduce the value of the numerator In the case of exponents, the effect is to reduce the power of the numerator Thus, a negative exponent represents an exponential in the denominator Ex: 3 AP Calculus AB, SJHS

E XPONENT A LGEBRA Because the exponent is the sum of times the base is multiplied by itself, multiplication of two exponentials of the same base results in addition of their exponents (with the same base) Multiplication and division are really the same operation, and this is made obvious since exponents in a denominator can be represented as negative numbers Thus, the result of division of exponentials of the same base is the subtraction of the numerator exponent by the denominator exponent (more generally, it is subtraction of the exponent you are moving on the fraction) Exponentials may also be raised to a power; this is the same as rewriting (and multiplying) that exponential as many times as its outer exponent suggests Thus, the results of raising an exponential to an exponent is to take the same base raised to a power that is the product of those exponents 4 AP Calculus AB, SJHS

S UMMARY OF E XPONENT A LGEBRA 5 AP Calculus AB, SJHS

E XAMPLES : S IMPLIFY IN TERMS OF ONLY POSITIVE EXPONENTS a) b) 6 AP Calculus AB, SJHS

T HE E XPONENTIAL F UNCTION The exponential, represented as e, is a number resulting from a limit most commonly seen in the finance problem of compound interest (we’ll take a look at this later) The number e is most simply approximated by: We say approximate since it is an irrational repeated decimal, like the number pi The exponential function, then, is simply the function f(x)=e x 7 AP Calculus AB, SJHS

L OGARITHMS The logarithm (or “log”) is the inverse operation & inverse function of the exponential For a general logarithmic equation, it is best to view the parts in terms of exponents: 8 AP Calculus AB, SJHS

S IMPLE EXAMPLES OF LOGARITHMS a) b) c) 9 AP Calculus AB, SJHS

S PECIAL L OGARITHMS Common Log Natural Log In everyday math speak, the common log is known as “common log” or “log ten”; the natural log is commonly said as “log”, “log n”, or “lan” So from here on, if no other designation is given, the spoken word “log” is assumed to mean base e, even if the written word log assumes a base AP Calculus AB, SJHS

S UMMARY OF L OGARITHM O PERATIONS 11 AP Calculus AB, SJHS

I NTERPRETATION OF L OGARITHMIC VS E XPONENTIAL O PERATIONS The best way to understand logarithmic operations is compare them side-by-side with exponential operations You’ll notice that in both instances, addition is linked to multiplication, subtraction is linked to division, and exponents are linked to multiplication 12 AP Calculus AB, SJHS

O THER IMPORTANT PROPERTIES OF LOGARITHMS 13 AP Calculus AB, SJHS

S IMPLIFYING L OGARITHMS To simplify a logarithm, the easiest method is to separate out the denominator, then any multiplications, then exponents Do look out for the following log forms that cannot be simplified: log (A+B) or log (A-B) log A / log B (logA) n or log n A 14 AP Calculus AB, SJHS

E XAMPLES : S IMPLIFY EACH LOGARITHM AS A SUM OF LOGARITHMS a) b) c) d) 15 AP Calculus AB, SJHS

E XPONENTIAL AND L OGARITHMIC E QUATIONS Because they are inverse functions, exponents and logarithms can be used to solve equations In order to solve, the equation must be in the form: Exponential Equation: Logarithmic Equation: To get your equation into these forms, you must: Isolate the exponential for the exponential equation Combine everything possible into the log (and isolate) for the logarithmic equation Note: many of these types of equations require a calculator, especially if they use the number e. 16 AP Calculus AB, SJHS

S OLVING L OGARITHMIC E QUATIONS USING THE I NVERSE P ROPERTY Some simple examples are: a) b) 17 AP Calculus AB, SJHS

E XAMPLES OF L OGARITHMIC E QUATIONS a) b) c) d) 18 AP Calculus AB, SJHS

A)A) 19 AP Calculus AB, SJHS

B)B) 20 AP Calculus AB, SJHS

C)C) 21 AP Calculus AB, SJHS

D)D) 22 AP Calculus AB, SJHS

S OLVING E XPONENTIAL E QUATIONS USING THE I NVERSE P ROPERTY Some simple examples are: a) b) 23 AP Calculus AB, SJHS

E XAMPLES OF E XPONENTIAL E QUATIONS a) b) c) 24 AP Calculus AB, SJHS

A)A) 25 AP Calculus AB, SJHS

B)B) 26 AP Calculus AB, SJHS

C)C) 27 AP Calculus AB, SJHS

C OMPOUND I NTEREST The best example of exponent use is in compound interest problems The idea of compound interest is to take an initial amount, known as the “principal”, and multiply it by some factor every compounding period This factor is often a number very close to 1, which ensures that the principal changes over time, but not too quickly 28 AP Calculus AB, SJHS

T HE C OMPOUND I NTEREST FORMULA P = present value P 0 = principal (initial amount) r = interest rate (annual) n = number of compounding periods t = time (years) For our purposes here, let’s consider the case of annually compounded interest, where there is only one compounding period in a year (n=1) 29 AP Calculus AB, SJHS

T HE C ONTINUOUSLY C OMPOUNDED I NTEREST F ORMULA An alternative is to compound interest continuously This means interest is always accumulating, every day, minute, second, and fraction of a second To do this, we would allow our number of compound periods in a year to approach infinity (remember limit notation?) As it turns out, the result of this limit uses the number e If we plug it in, we get our continuous compounding formula 30 AP Calculus AB, SJHS

W HICH IS B ETTER ? It depends on your objective! For simplicity, many banks compound savings interest monthly For a CD or other investment, your compounding period may be 1, 2, 5, 10, or even 20 years (or more!) So for example, a 10% interest investment of $1000 for 5 years will be worth $1000 in years 0, 1, 2, 3, and 4. Only in the 5 year will it be worth $ $(1000)(.10) = $1100 Credit cards often compound daily, so you owe more each day you put off paying, if only by a few cents Some loans compound continuously, so you might owe more on a student loan in the afternoon than you did a few hours earlier again, usually a small difference like a few cents, but it adds up 31 AP Calculus AB, SJHS

A COMPARISON OF INITIAL AMOUNTS You wish to grow a principal investment up to $ over 10 years at an interest rate of 14.2%. How much should you invest if the interest is compounded: (1) Annually & (2) Continuously Here, we have P 0 =?, P=10000, r=.142, and t=10. At this point, we can just plug it in to each case! Annual: Continuous: 32 AP Calculus AB, SJHS

A COMPARISON OF TIMES We want to grow an initial investment P 0 =5000 to a final amount P= Given an interest rate of 9.25%, how long will this take for: (1) Annual &(2) Continuous compounding 33 AP Calculus AB, SJHS

A NNUAL : AP Calculus AB, SJHS

C ONTINUOUS : AP Calculus AB, SJHS

E ND AP Calculus AB, SJHS