Logarithmic Units (§5.3)

1. Logarithmic Units (§5.3) It’s natural to measure things that grow exponentially by the logarithm of the actual value because log x grows linearly as x grows exponentially. Example: pH Chemical concentrations tend to increase or decrease exponentially with changes like temperature or the presence of catalysts. So it makes sense to define the alkalinity of a solution by the formula pH = -log10([H30+]). In this way, an exponential decrease in hydronium ion concentration leads to a linear increase in pH; i.e., the solution becomes more alkaline. Example: dB Electrical amplifiers affect intensities like sound levels and power by multiplying by “gains”. By measuring in logarithmic units, these multiplications become addition. If I0 is a base intensity level which has been amplified to the level I then 10log10(I/I0) is the intensity gain measured in decibels (dB). 1 dB is a gain of 10(1/10) ≈ 1.26 so a 1dB gain multiplies I0 by 1.26, a -1dB gain (i.e., a loss) divides I0 by 1.26. 3 dB ≈ 2.00 so a ±3db gain means multiplying or dividing I0 by 2. 10 dB = 1 Bel (named after Alexander Graham Bell) which corresponds to multiplying by a factor of 10. Example: The Richter Scale The intensity of an earthquake is measured as log10(E/E0) where E0 is the intensity of a very small earthquake. So an earthquake that’s 8.0 on the Richter scale is 100,000,000 times E0 and is catastrophic.

2. §5.6 Applications of Logarithms 4A0 A0 t = 2td t = 0 A0 / 2 2A0 A0 / 4 t = th A0 t = td A0 / 8 t = 0 t = 2th t = 3th A(t) = A0 2(t/td), td = Doubling Time A(t) = A0 (1/2)(t/th), th = Half-life Can also be written as A(t) = A0ekt Can also be written as A(t) = A0e-kt Don’t memorize these formulas: Just understand how to convert from one form to the other.