Solving Logarithmic equations

Solving Logarithmic equationsNicholas SchleyerHistoryLogarithms were invented by John Napier of Scotland and Joost Burgi of Switzerland. They wanted to find a way to simplify mathematical calculations.They worked independently of each other; they never met.Their inventions were very different from each other.In 1614, Napier published his definition of logarithms as a ratio of two distances in a geometric form.Logarithms were not defined as exponents until John Wallis in 1685 and Johann Bernoulli in 1694.

SolvingRewrite the equation in exponential form and solve for the variable.Properties of LogsWhen solving, it can also be useful to know some of the properties of logarithms:Logb(mn) = logb(m) + logb(n) Product PropertyLogb(m/n) = logb(m) logb(b) Quotient Propertylogb(mn) = n * logb(m) Power PropertyLogb(b)x = x Exponent Logarithm Inverse Property 1Blogbx = x (iff x > 0) Exponent Logarithm Inverse Property 2If Logb(X) = logb(Y), then x = y One to One PropertyLn(mn) = ln(m) + ln(n)Ln(m/n) = ln(m) ln(n)Ln(mn) = n * ln(m)Example 17Log(3x) = 15Example 17Log(3x) = 15Isolate logarithmic term, divide both sides by 7Log(3x) = 15/7Convert to exponential equation 10^(15/7) = 3xDivide both sides by 3X = (10^(15/7))/3 = 46.3Example 22logb(x) = logb(4) + logb(x-1)2logb(X) = logb(4) + logb(x-1)Logb(x2) = logb((4)(x-1)) Power PropertyLogb(x2) = logb(4x-4) FoilX2 = 4x 4 One to OneX2 4x + 4 = 0(x-2)(x-2) = 0 FactorX = 2

Example 3Ln(ex) = ln(e3) + ln(e5)Example 3Ln(ex) = ln(e3) + ln(e5)Ln(ex) = ln((e3)(e5)) Product PropertyLn(ex) = ln(e3+5)Ln(ex) = ln(e8)ex = e8 One to OneX = 8

SOurceshttp://www.purplemath.com/modules/solvelog.htmhttp://www.sosmath.com/algebra/logs/log4/log47/log47.html