Introduction to Logarithms

• The logarithm of a number is the exponent to which the base must be raised to produce that number.
• Example: How many \(3\)’s do we multiply to get \(81\)? Answer:\(3 \times 3 \times 3 \times 3 = 3^4 = 81\). So the logarithm to the base \(3\) of \(81\) is \(4\).
• We write "the number of \(3\)’s you need to multiply to get \(81\) is \(4\)" as \(\log_{3}(81)=4\). So these two things are equivalent:

\[ 3 \times 3 \times 3\times 3= 3^4= 81 \quad\Longleftrightarrow\quad \log_{3} (81) = 4 \]

• The number we are multiplying is called the "base", so we would say: "the logarithm of 81 with base 3 is 4" or "logarithm of 81, to the base 3, is 4".
• For example: what is \(\log_5(625)\)? In this case we are asking, "how many \(5\)’s need to be multiplied together to get \(625\)?"

\[ 5 \times 5 \times 5 \times 5 = 5^{4} = 625\]

  so we need \(4\) lots of \(5\)’s. Therefore, the answer is:

\[ \log_{5}(625) = 4 \]

• We write the logarithm to the base \(10\) as \(\log_{10} = \log\).
• There is a special case for logarithms, when we use the irrational number, \(e\approx2.718281\ldots\) as the base. When using this base, \(\log_e\) we say the natural logarithm and can use the shorter notation of \(\log_e=\ln\).