Expert: Sherry Wallin - 5/31/2009

Question
QUESTION: how do you evaluate log(3)243

ANSWER: Hi Devin~
    You are really looking for what power to raise the base to to get the argument. In other words, in your problem, you are looking for what power to raise 3 to in order to get 243.  3^5 = 243

Math Prof

---------- FOLLOW-UP ----------

QUESTION: if there are initially 2500 bacteria in a culture and the number of bacteria double each hour.how long would it take the culture to grow 125000 bacteria?

Answer
Hi Devin~
    Please feel free to ask new questions but don't do them as follow up questions because this question will be cataloged as a log problem and not an exponential problem, ok?
   When something doubles it means it is multiplied by 2. So after the first hour you would have 2(2500) = 5000 right? After the 2nd hour you would have 2*5000 = 2*2*2500 = 10,000, right? And after the 3rd hour you would have 2*10,000 = 2*2*2*2500 = 20,000, yes? Which is equal to 2^3*2500. Notice the 3rd power 2 is raised to matches the fact that we want to know how many bacteria we have after 3 hours and likewise for 2 hours and 1 hour. It stands to reason that there is a pattern here that can be thought of as the number of bacteria after n hours is going to  be 2^n*2500. So we want to know when that is 125,000. So set the equation up and solve for n. 2^n*2500 = 125000 divide both side by 2500 getting 2^n = 50. You need a calculator to solve this exactly but you know that 2^5 = 32 and 2^6 = 64 and 50 is somewhere in between 32 and 64 so it must be that between 5 and 6 hours we reach 125,000 bacteria so the answer is 6 hours. Note if 50 could be written as a power of 2 you would rewrite 50 as 2^(some power) = 2^n and then n would equal that some power.

Math Prof