Properties of Logs (6.10)

Properties of Logs (6.10) Three useful tools to solve Logarithmic equations

Problem o’ the Day Solve for x: logx (1/64) = 2

Problem o’ the Day Solve for x: logx (1/64) = 2 x2 = 1/64 x = 1/8

Property #1 logb(xy) = logb x + logb y Notice how we can split out the logs of x and y. Compare these logs: 1. log 15, log 3 + log 5 2. log 50, log 2 + log 25, log 10 + log 5

A small aside Out of curiosity, is log 5 a rational or irrational number? Try it: punch log 5 into your calculator, and when you get the answer, hit Math-Frac.

A small aside Out of curiosity, is the log 5 a rational or irrational number? Try it: punch log 5 into your calculator, and when you get the answer, hit Math-Frac. You don’t get a fraction, do you? That’s because log 5 can’t be expressed as a fraction of integers. That means log 5 is an irrational number!

Property #2 logb (x/y) = logb x - logby Like the last one, only division and subtraction are matched up. Compare these logs: 1. log 15, log 75 - log 5 2. log 50, log 100 - log 2, log 200 - log 4

Caution! Be careful about how you use these. logbx + logby ≠ logb(x + y) logbx - logby ≠ logb(x - y) You cannot distribute the log!

Property #3 You might have seen this before briefly. logb xy = y(logb x) In this one, you can bring the exponent down. Compare these logs: 1. log (53), 3 log 5

Clarification The following comparisons may help clarify logarithmic relationships with exponents. log 24 = 4(log 2) = log 2 + log 2 + log 2 + log 2 (log2)4 = (log 2)(log 2)(log 2)(log 2) Notice how they are not the same thing!

Try them Given log 2 = .301 log 3 = .477 • log 6 • log 4 • log 12 • log 48

Try them Given log 2 = .301 log 3 = .477 • log 6 = .301 + .477 = .778 • log 4 = .301 + .301 = .602 • log 12 = .301 + .301 + .477 = 1.079 • log 48 = 4(.301) + .477 = 1.681 6

Try them 1. Combine into a single argument: log 3 + 2 log5 2. Solve for x: 7x = 83 There are two ways to do this one. a. Log each side, then bring the x down and solve. b. Rewrite the equation for x and use the change of base.

Try them 1. Combine into a single argument: log 3 + 2 log5 = log 3 + 2 log 5 = log(3*25) = log (75) 2. Solve for x: The two ways to do this one. a. Log each side, then bring the x down and solve. 7x = 83 log 7x = log 83 x(log 7) = log 83; x = (log 83)/(log 7) = 2.271 b. Rewrite the equation for x and use the change of base. x = log7 83 x = log (83)/log(7) = 2.271

Properties of Logs (6.10)