Review of Logarithms Review of Logarithms On Exam

Review of Logarithms

Review of Logarithms • On Exam I: By some questions I received, I think that some of you have an appalling lack of understanding of some basic rules about how to use logarithms! • Example: Using Ω 1(E 1) = K(E 1)f 1, etc. calculate the ratio: R = [Ω 1(E 1 = 3. 15)Ω 2(E 2 = 4. 7)]/[Ω 1(E 1 = 3. 14)Ω 2(E 2 = 4. 69)] or R = (Numerator)/(Denominator) • Some of you wrote ln(R) = ln(Numerator)/ln(Denominator) which is 100% nonsensical!! • The correct expression is ln(R) = ln(Numerator) – ln(Denominator) • Also other very basic errors!!

Rules of Logarithms If M & N are positive real numbers & b ≠ 1: • Product Rule: logb. MN = logb. M + logb. N The log of a product equals the sum of the logs Examples: log 4(7 • 9) = log 47 + log 49 log (10 x) = log 10 + log x log 8(13 • 9) = log 8(13) + log 8(9) log 7(1000 x) = log 7(1000) + log 7(x)

Rules of Logarithms If M & N are positive real numbers & b ≠ 1: • Quotient Rule: logb(M/N) = logb. M - logb. N The log of a quotient equals the difference of the logs Example: log[(½)x] = log(x) + log(½) = log(x) + log(1) – log(2) But, log (1) = 0, so log[(½)x] = log(x) - log(2)

Rules of Logarithms If M & N are positive real numbers & b ≠ 1 & p is any real number: • Power Rule: logb. Mp = p logb. M The log of a number with an exponent equals the product of the exponent & the log of that number Examples: log x 2 = 2 log x ln 74 = 4 ln 7 log 359 = 9 log 35

Change of Base Formula • Most often we use either base 10 or base e • Most calculators have the ability to do either. • How can we use a calculator to compute the log of a number when the base is neither 10 nor e? • Use the formula Example: log 2 (17) = ? • So, log 2 (17) = [logb(x)/logb(a)]

Basic Properties of Logarithms • Most used properties:

Using the Log Function for Solutions Example • Solve for t: • Take the log of both sides & use properties of logs

Properties of the Natural Logarithm • Recall that y = ln x x = ey • Note that – ln 1 = 0 and ln e = 1 – ln (ex) = x (for all x) – e ln x = x (for x > 0) • As with other based logarithms

Use Properties for Solving Exponential Equations • Given • Take log of both sides • Use exponent property • Solve for what was the exponent Note this is not the same as log 1. 04 – log 3

Common Errors & Misconceptions log (a+b) is NOT the same as log a + log b log (a-b) is NOT the same as log a – log b log (a*b) is NOT the same as (log a)(log b) log (a/b) is NOT the same as (log a)/(log b) log (1/a) is NOT the same as 1/(log a)