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Copenagle, Academic SupportPage 1/6What is a logarithm?•To answer this, first try to answer the following:what is x in this equation? 9 = 3xwhat is x in this equation? 8 = 2x•Basically, logarithmic transformations ask, “a number, to whatpower equals another number?”•In particular, logs do that for specific numbers under the exponent. This number is called the base.•In your classes you will really only encounter logs for two bases, 10 and e.Log base 10We write “log base ten” as “log10” or just “log” for short and we define it like this:Ify = 10xthenlog (y) = xSo, what is log (10x) ?log (10x) = xHow about 10log(x) ?10log(x) = xMore examples:log 100 = 2log (105)=5•The point starts to emerge that logs are really shorthand for exponents. •Logs were invented to turn multiplication problems into addition problems. Lets see why.log (102) + log (103) = 5, or log (105)
Copenagle, Academic SupportPage 2/6•So, clearly there’s a parallel between the rules of exponents and the rules of logs:Table 4. Functions of log base 10.ExponentsLog base 10Examplessrsraaa+=!log(AB) = log(A) + log(B)log(105) = log (102) + log (103)ssaa!=1log !"#$%&B1= - log(B)log!"#$%&5101= log (10-5)= -log(105)srsraaa!=log!"#$%&BA= log(A) – log(B)log(102) = log (105) - log (103) = 5 – 3 = 2rssraa=)(log (Ax) = xlog(A)log(103) = 3log(10) = 3 (1) = 310=alog(1) = 0log(10) = 1
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