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MATH Resources for Academic Success NFCC Exponential and Logarithmic Study Guide w/links Before starting this overview, please review the rules of exponents, as well as inverse functions. They are the foundation you need for this unit. Exponent rules review: https://www.khanacademy.org/math/algebra/exponent-equations/exponentproperties-algebra/v/negative-and-positive-exponents Exponential Functions (5.1 of Pearson College Algebra text) An exponential function looks like f(x) = bx. b must be a positive number (otherwise f(x) isn't a real number almost everywhere) and b can't be 1 (otherwise it's just 1x, which is always 1 so not very interesting). Exponential growth occurs when b>1, and exponential decay occurs when 0<b<1. To tell if a function exhibits exponential growth or decay, one easy way is to look at the graph. Many things in real life can be modeled by exponential functions. Population, for example, can exhibit exponential growth; the temperature of a hot object cooling to room temperature can exhibit exponential decay. We see this in the graph of the data. The constant value e shows up in many contexts in exponential functions. Just like pi is a constant that shows up in circles, e is useful enough to have its own name. e is approximately 2.71828... On your calculator, you can find e by hitting (2nd) (ln), it'll paste “e^( “ (The calculator helpfully gives you the exponent and start of parentheses, since e almost always is raised to some power). In math we agree to NEVER use e as a variable; it is always used to represent the number 2.71828... . Logarithmic Functions (5.2) A logarithmic function looks like f(x) = loga (x). (Sometimes they don't write parentheses around the x for logarithmic functions; trig functions don't have parentheses either. I try to always write the parentheses, to be clear that the logarithm is a function.) loga (x) is the number you have to raise a to the power of to get x. Notice that a is a subscript, and loga (x) is read "log base a of x," NOT "log of a to the x." The function f(x) = loga (x) is the inverse (ch. 4.3 in the Pearson College Algebra text) of the function f-1(x) = ax. So, since a came from the base of an exponential function, a has to be positive and not zero. (For example, f(x) = log-3 (x) isn't a function.) So, loga (x) = y MEANS ya=x; these are two ways to say the exact same thing: "the number you have to raise a to the power of to get x is y" and "y to the power of a is x" are the same sentence rearranged. This is for any base a that is positive and not 1) You can use the definition and rules to evaluate certain logs; for example, log3 (81) = 4, since 34 = 81. Because they are used so much, there is special notation for base 10 and base e: log (x) (no base written) means log10 (x) ln (x) (natural log) means loge (x) (the number e from 5.1) Developed by Glenn Bryce for NFCC, ed. 2015 MATH Resources for Academic Success We can get basic rules for exponents from the definition, as follows: loga (1) = 0 (since a0 = 1) loga (a) = 1 (since a1 = a) loga (ax) = x (inverse functions compose to the identity function—they “cancel” each other out) a(log a (x)) = x ( inverse functions compose to the identity function—they “cancel” each other out) Properties of Logarithms Recall that “logb x” is the power you raise b to in order to get x, so “logb x” is an exponent. Since it is an exponent, it follows the exponent rules. For example, when multiplying with like bases you would add the exponents. Take a look at this relationship, below. Exponent rule Logarithm rule aman = am+n loga(mn) = logam + logan mult powers = add exponents mult inside = add “exponents” outside (remember, logs ARE exponents) (am)/(an) = am-n loga(m/n) = logam - logan div powers = subt exponents divide inside = subt “exponents” outside (am)n = amn loga(mr) = r logam raise power to power = mult exponents exp inside = mult “exponents” outside http://www.purplemath.com/modules/logrules.htm You can use these rules to write logarithmic expressions as a sum of logs, or as a single log—just reverse the process by using the rules to convert "outside" operations to "inside" ones. Note, you can only do what the rules above say and no more. So for example, log(x+1) can't be written any other way, because there is no rule for addition of powers. For example, we cannot simplify x2 + x3 by adding exponents! http://www.youtube.com/watch?v=hyLPjvpjWdU and https://www.youtube.com/watch?v=-1x6E7E66ow Change of base formula Notice there's no “loga” button on your calculator, only “log” (implied base 10) and “ln” (implied base e). However, you can use the change of base formula to convert any base logarithm to any other base logarithm: logax = logb(x) / logb(a) But the formula is most useful if you let b = 10 or e because you have those buttons on your calculator: logax = log(x) / log(a) OR logax = ln(x) / ln(a) Developed by Glenn Bryce for NFCC, ed. 2015 MATH Resources for Academic Success Ex) Evaluate log5 8. log5 8 = ln(8)/ln(5) = 1.292... since 1.292 is the number you have to raise 5 to the power of to get 8. OR log5 8 = log(8)/log(5) = 1.292... the same answer since we can use ANY new base. https://www.khanacademy.org/math/algebra2/logarithms-tutorial Solving equations with exponents and logs Since the statements ax = y and loga(y) = x are equivalent, if we're solving an equation with our variable inside a logarithmic function, we can get our equation to look like loga(something) = (something else) and rewrite as an exponent to get the variable down from the exponent. Alternative approach to same problem*: Ex) Solve: 2 log3(x-1) = 4 log3(x-1) = 2 divided both sides by 2 to isolate x x - 1 = 32 rewrote as an exponent x-1=9 32 = 9 x = 10 added 1 to both sides to isolate x Ex) Solve: 2 log3(x-1) = 4 log3(x-1) = 2 divided both sides by 2 to isolate x 3 log3(x-1) = 32 used inverse base of 3 x - 1 = 32 inverse functions 3x and log3x cancelled x-1=9 32 = 9 x = 10 added 1 to both sides to isolate x NOTE: If our variable starts out inside a logarithm, we MUST check our answer at the end because of restrictions on the domain. The book’s author forgets to mention this. Remember that the domain of a logarithmic function is limited to the positive real numbers, so if we substitute in order to check our answer and it shows we have to take a logarithm of a negative number, that answer isn't a solution after all. To get variables out of exponents, we can take any base logarithm of both sides and use the property exponent inside <-> multiplication outside to bring the exponent down. Since it doesn't matter what base log we take, we might as well use one our calculator can do, i.e. e or 10. Alternative approach to same problem**: x+1 Ex) Solve: 1 + 5 = 10 5x+1 = 9 subtracted 1 from both sides to isolate x ln(5x+1)= ln(9) natural log of both sides; could have used log base10 (x+1) ln 5 = ln 9 properties of logs x+1 = (ln 9)/(ln 5) divide by ln 5 to isolate x x = (ln 9)/(ln 5) - 1 subtract 1 from both sides to isolate x x = 0.3652... used calculator Ex) Solve: 1 + 5x+1 = 10 5x+1 = 9 subtracted 1 from both sides to isolate x log5(5x+1)= log5(9) used inverse log5 x+1 = log59 inverse functions log5x and 5x cancelled x = log59 - 1 subtracted 1 from both sides to isolate x x = log5/log9 - 1 used change of bases formula x = 0.3652... used calculator We don't have to check our solutions here, since exponential functions are defined everywhere. https://www.khanacademy.org/math/algebra2/exponential_and_logarithmic_func/logarithmicequations/v/solving-logarithmic-equations *Note alternative approach at 3:41 https://www.khanacademy.org/math/algebra2/exponential_and_logarithmic_func/exponentialmodeling/v/solve-exponentials **Alternative approach used in this video https://www.youtube.com/watch?v=AlWK-VYE7Hw lots of examples Developed by Glenn Bryce for NFCC, ed. 2015