Logarithms

A review of the definition of the logarithm and logarithm properties

The Essentials

A logarithm is the ’opposite’ of an exponential:

\[ y = \log_b(x) \]

this is read as the log base b of x is equal to y. It is an equivalent expression to:

\[ x = b^y \]

For example the expression \( \log_{5}(125) \) is equal to 3 because \( 5^3 = 125 \).

If a logarithm doesn’t have a specified base, then it is of base ten: \( \log(100) \equiv \log_{10}(100) \). Another special base for logarithms is base e. The number \( e \) is approximately \( e \approx 2.7182818 \), and like pi, it is a very significant number found in nature. The base e log is called a ’natural logarithm’ or a ’natural log’ and is written as \( \log_e(x) = \ln(x) \).

Some properties of logarithms include:

\[ \log_b b = 1 \]

(Since \( b^1 = b \) )

\[ \log_b 1 = 0 \]

(Since \( b^0 = 1) \)

\[ \log_b (b^x) = x \]
\[ b^{\log_b(x)} = x \]
\[ \log_b(x^r) = r \log_b(x) \]
\[ \log_b(xy) = \log_b(x) + \log_b(y) \]
\[ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \]

The domain of any logarithm is \( x > 0 \) and the \( \lim_{{x \to 0}} \log(x) = -\infty \). The base of a logarithm can be changed using this formula:

\[ \log_b(a) = \frac{\log_x(b)}{\log_x(a)} \]

Example

Evaluate this expression:

\[ \log_5(b) \cdot \log_b(25) \]

Using the change of base formula we can re-write both of these logs as base 10 logs:

\[ \frac{\log(b)}{\log(5)} \cdot \frac{\log(25)}{\log(b)} \] \[ \frac{\log(25)}{\log(5)} \]

Using the change of base formula again we can re-write the expression as:

\[ \log_5(25) = 2 \]

Practice

Evaluate these expressions:

  1. \( \log_6(216) \)
  2. \( \log_3(50) \cdot \log_{50}(27) \)
  3. \( \log_{17}(1) \)

Solve these equation for \( y(x) \):

  1. \( 5 \ln\left(\frac{5}{y}\right) = x \)
  2. \( \frac{\log\left( 8/y \right)}{4} = \frac{3}{4}x + \log\left(\frac{1}{y}\right) \)

Solution:

  1. \( 3 \)
  2. \( 3 \)
  3. \( 0 \)
  4. \( y(x) = 5e^{-x/5} \)
  5. \(y(x) = \frac{10^x}{2} \)

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