REVIEW OF LOGS AND EXPONENTIALS AND POWERS ---------Warren D Smith April 2003---------------- LOGARITHMS (LOGS) ================= You can take the log of any POSITIVE number. The most important property of logs is log(x)+log(y)=log(xy) ; MULTIPLYING two numbers and taking the log is the same as ADDING their logs. (Reason logs were invented: to save labor by replacing multiplies with adds, allowing slide rules to work, etc.) Other properties (all of which follow from the above master property - if you figure out how each one follows, that will make all these peanuts to remember): POWERS: P log(a ) = P log(a). Examples with P=2, 3, -1, and -2: 2 3 1 1 log(a ) = 2 log(a) log(a ) = 3 log(a), log(---) = -log(a), log(---) = -2 log(a) a 2 a DIVISION: log(A/B) = log(A) - log(B) POWERS OF TEN (for "common logs", i.e. logs to the base 10): P x log(M) log(10 ) = P <--- log and 10 undo each other: also 10 = M. antilog examples with P=0,1,2,3,4,-1,-2,-3,-4: log(1) = 0, log(10) = 1, log(100) = 2, log(1000) = 3, log(10000) = 4, log(.1) = -1, log(.01) = -2, log(.001) = -3, log(.0001) = -4 log(57) 574 10 = 57 log(10 ) = 574 POWERS & EXPONENTIALS ===================== B A = A A A A A ... A <--B in a row, all multiplied together if B>0. -B A = 1 / (A A A A A ... A) <--B in a row, all multiplied together. So note: negative powers are got by 1/(the positive power). P 10 = 10000000 <-- P zeros -P 10 = 0.000001 <-- P-1 zeros after the decimal point Examples: 0 1 2 3 A = 1 A = A A = AA A = AAA -1 -2 -3 A = 1/A A = 1/(AA) A = 1/(AAA). PROPERTIES: BC B C A = (A ) <-- a power of a power: same as multiply the two powers B C B+C A A = A B C B-C A / A = A B 1/B (A ) = A HOW TO SOLVE EQUATIONS INVOLVING LOGS, EXPONENTIALS, POWERS: ============================================================ X A = B <-- solve for X. Answer (by taking logs of both sides, using properties above) X = log(B) / log(A). A X = B <-- solve for X. Answer (by taking 1/A power of both sides, using properties above) 1/A X = B EXPONENTIAL GROWTH, HALFLIVES, DOUBLING TIMES, AND ALL THAT: ============================================================ A quantity is growing LINEARLY if you keep adding some fixed increment to it every time slot. A quantity is decaying LINEARLY if you keep subtracting some fixed decrement from it every time slot. Law: y = Slope * X + Yintercept Slope = rise/run, Yintercept = where line crosses Yaxis A quantity is growing EXPONENTIALLY if you keep multiplying it by some fixed factor c>1 every time slot. A quantity is decaying EXPONENTIALLY if you keep multiplying it by some fixed factor c (01, growth) and similarly if c<1 we have Thalving = -log(2)/log(c) (if c<1, decay). You can always solve these things by using logs and powers and their properties. EXAMPLES OF SOLVED PROBLEMS: Example: -77/4321. halflife = 4321 years. So Fraction left after 77 years = 2 Example: fraction left after 345 years is 0.12. What is halflife? ANS: -345/Halflife 2 = 0.12 take logs (-345/Halflife) log(2) = log(0.12) solve -345 * log(2) / log(0.12) = Halflife = 112.79. Example: stuff grows 3% per year. How many years Y before increase by factor of 10? ANS: Y (1.03) = 10 take logs Y log(1.03) = log(10) solve Y = log(10)/log(1.03) = 1/log(1.03) = 77.90. Example: Stuff increases by a factor of C each year. In 105 years it has increased by a factor of 1732. What is C? What is percent increase per year? ANS: 105 C = 1732 take power 1/105 of both sides and use properties: 105*(1/105) 1 1/105 C = C = C = 1732 = 1.073602003. And so it's 7.36% increase per year. -------------------------------------------------