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 Ω1(E1) = K(E1)f1, etc. calculate the ratio: R = [Ω1(E1 = 3.15)Ω2(E2 = 4.7)]/[Ω1(E1 = 3.14)Ω2(E2 = 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: logbMN = logbM + logbN The log of a product equals the sum of the logs Examples: log4(7 • 9) = log47 + log49 log (10x) = log10 + log x log8(13 • 9) = log8(13) + log8(9) log7(1000x) = log7(1000) + log7(x)

Rules of Logarithms If M & N are positive real numbers & b ≠ 1: Quotient Rule: logb(M/N) = logbM - logbN 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: logbMp = p logbM The log of a number with an exponent equals the product of the exponent & the log of that number Examples: log x2 = 2 log x ln 74 = 4 ln 7 log359 = 9log35

log2 (17) = [logb(x)/logb(a)] 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? Example: log2 (17) = ? Use the formula So, log2 (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)