Logarithms and their properties
A logarithm answers the question 'what power?'. Logarithms undo exponentials and turn multiplication into addition, which made them a calculating tool for centuries.
Definition
The logarithm base b of x is the exponent you put on b to get x: \(\log_b x = y\) means \(b^y = x\). So \(\log_2 8 = 3\) because \(2^3 = 8\), and \(\log_{10} 0.01 = -2\) because \(10^{-2} = 0.01\). Logarithms are defined only for positive x and a base b > 0, b not 1.
Common and natural logs
The common logarithm \(\log x\) uses base 10, and the natural logarithm \(\ln x\) uses base e. So \(\log 1000 = 3\) and \(\ln e^5 = 5\). Logarithms were introduced by John Napier in 1614 to simplify astronomical calculations.
Inverse of the exponential
The functions \(y = b^x\) and \(y = \log_b x\) are inverses, so \(\log_b(b^x) = x\) and \(b^{\log_b x} = x\). The graph of \(y = \log_b x\) is the reflection of \(y = b^x\) across \(y = x\): it passes through (1, 0), has domain x > 0, and has a vertical asymptote at x = 0.
Product and quotient rules
Because exponents add when powers multiply, logs turn products into sums: \(\log_b(MN) = \log_b M + \log_b N\). Similarly, \(\log_b\frac{M}{N} = \log_b M - \log_b N\). For example, \(\log 2 + \log 50 = \log 100 = 2\).
Power rule
\(\log_b(M^p) = p\log_b M\). This rule brings exponents down, which is the key step in solving exponential equations: \(\log 10^7 = 7\), and \(\ln x^3 = 3\ln x\). There is no rule for the log of a sum: \(\log(M + N)\) is not \(\log M + \log N\).
Change of base
To evaluate logs in any base with a calculator, use \(\log_b x = \frac{\ln x}{\ln b}\) or \(\frac{\log x}{\log b}\). So \(\log_2 50 = \frac{\ln 50}{\ln 2} \approx \frac{3.912}{0.693} \approx 5.64\), meaning \(2^{5.64} \approx 50\).
Logarithmic scales
Logarithms compress huge ranges into small numbers. The pH scale is pH = -log of the hydrogen ion concentration: lemon juice at pH 2 is ten times more acidic than pH 3. Decibels and earthquake magnitude scales are also logarithmic; each whole step in earthquake magnitude means about 32 times more energy released.
Worked example
Expand \(\log_3\frac{9x^2}{y}\). Quotient rule: \(\log_3(9x^2) - \log_3 y\). Product rule: \(\log_3 9 + \log_3 x^2 - \log_3 y\). Power rule and \(\log_3 9 = 2\): the result is \(2 + 2\log_3 x - \log_3 y\).
Key takeaways
log base b of x is the exponent that turns b into x; logs need positive inputs. Common logs use base 10 and natural logs base e; Napier introduced logarithms in 1614. Logs and exponentials are inverses, so their graphs reflect across y = x, and log graphs pass through (1, 0) with a vertical asymptote at x = 0. The product, quotient, and power rules turn multiplication into addition, division into subtraction, and exponents into multipliers; there is no rule for the log of a sum. Change of base allows calculator evaluation. pH, decibels, and earthquake magnitudes are logarithmic scales.
Flashcard
log_b x = y means b^y = x
Check your understanding
What is \(\log_2 8\)?