Logarithms

Definitionlogarithm50pix

Logarithm is a fancy (and confusing) word for exponent.  Actually, if you know how to read it from left to right, then it’s not so confusing. When a logarithm is written, at least two pieces of information are given: the “base”, and the derived value.  An example:

\(\begin{align} \log_3 9 = 2\end{align}\)

In this example, 3 is the base, and 9 is the derived value. Translated, this should be read from left to right as: “The exponent you give 3 to get 9 is 2.” Or, in other words, \(\begin{align}3^2 = 9\end{align}\). In more general terms, “The exponent you give the base (3 in this example) to get the value (9 in this example) is \(\begin{align}\log_3 9\end{align}\). Since we know that \(\begin{align}3^2 = 9\end{align}\), then \(\begin{align}\log_3 9\end{align}\) is 2. More examples:

    • \(\begin{align} \log_5 125 = 3\end{align}\) because \(\begin{align} 5^3 = 125\end{align}\)
    • \(\begin{align} \log_{25} 5 = \frac{1}{2}\end{align}\) because \(\begin{align}25^\frac{1}{2} = \sqrt{25} = 5\end{align}\)
    • \(\begin{align} \log_4 \frac{1}{16} = -2\end{align}\) because \(\begin{align}4^{-2} = \frac{1}{4^2} = \frac{1}{16}\end{align}\)

In general, if \(\begin{align} a^b = c\end{align}\) then \(\begin{align} \log_a c = b\end{align}\)

And more examples

  • \(\begin{align} \log_5 5 = 1\end{align}\) because \(\begin{align} 5^1 = 5\end{align}\)
  • \(\begin{align} \log_5 25 = 2\end{align}\) because \(\begin{align} 5^2 = 25\end{align}\)
  • \(\begin{align} \log_5 625 = 4\end{align}\) because \(\begin{align} 5^4 = 625\end{align}\)
  • etc.
  • \(\begin{align} \log_{125} 5 = \frac{1}{3}\end{align}\) because \(\begin{align}5 \times 5 \times 5 = 125\end{align}\) so \(\begin{align}125^\frac{1}{3} = \sqrt[3]{125} = 5\end{align}\)
  • \(\begin{align} \log_{100} 1 =0\end{align}\) because \(\begin{align}100^0 = 1\end{align}\)
  • \(\begin{align} \log_{125} 1 = 0\end{align}\) because \(\begin{align}125^0 = 1\end{align}\)
  • \(\begin{align} \log_4 \frac{1}{4} = -1\end{align}\) because \(\begin{align}4^{-1} = \frac{1}{4}\end{align}\)
  • \(\begin{align} \log_4 \frac{1}{64} = -3\end{align}\) because \(\begin{align}4^{-3} = \frac{1}{4^3} = \frac{1}{64}\end{align}\)
  • etc.
  • \(\begin{align} \log_8 4 = \frac{2}{3}\end{align}\) because \(\begin{align}8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4\end{align}\)

See also Exponents