Lesson 5 of 9, 12 min read
Properties of logarithms
The product, quotient and power rules, expanding and condensing logarithmic expressions, the change-of-base formula and logarithmic scales like pH.
From exponent rules to log rules
A logarithm is an exponent, so every exponent rule has a logarithm version. Multiplying powers adds exponents; so the log of a product is the sum of the logs.
Expanding and condensing
Expanding writes one logarithm as a sum and difference of simpler ones. Condensing does the reverse.
Change of base
Calculators only have \(\log\) and \(\ln\). To evaluate \(\log_3 20\):
Logarithmic scales
Some quantities range over many powers of ten, so scientists measure them with logarithms. The pH of a solution is
where \([\text{H}^+]\) is the hydrogen ion concentration. Lemon juice with \([\text{H}^+] = 10^{-2}\) has pH \(2\); pure water with \(10^{-7}\) has pH \(7\). Because of the logarithm, each drop of \(1\) in pH means the solution is ten times more acidic. The Richter scale for earthquakes and the decibel scale for sound work the same way.
Practice
Quick check
Which expression equals \(\log_b M^k\)?
That's the power rule.
Quick check
Which expression equals \(\log(10x)\), for \(x > 0\)?
Product rule: \(\log 10 + \log x = 1 + \log x\).
Finished this lesson?
Mark it as completed and it's saved on this device. Sign up to see it everywhere.