Logarithms in Physics — Decibels, Half-Life and the Richter Scale

What a logarithm is asking

$\log_{10}(1000) = 3$ because $10^3 = 1000$. The logarithm is the exponent — it asks 'base raised to what power?' Two bases dominate physics: base 10 ($\log_{10}$, for human-scale ratios like decibels) and base $e \approx 2.718$ ($\ln$, the natural log, for anything involving growth and decay).

The two differ only by a constant factor: $\ln x = \ln(10)\,\log_{10} x \approx 2.303\,\log_{10} x$. Nature prefers $e$ because $d/dx\,e^x = e^x$ — the exponential is its own derivative, which makes $\ln$ the natural language of differential equations.

The three log laws

Everything you will ever need follows from three rules, true for any base:

• Product: $\log(ab) = \log a + \log b$
• Quotient: $\log(a/b) = \log a - \log b$
• Power: $\log(a^n) = n\,\log a$

Logs turn multiplication into addition — the trick behind slide rules and behind every log scale in science. Example: $\log_{10}(2 \times 10^5) = \log_{10} 2 + 5 \approx 5.30$. A factor of two is just $+0.30$ on the log scale.

Decibels: the sound scale

Sound intensity $I$ (power per area) runs from $10^{-12}\ \mathrm{W/m^2}$ at the hearing threshold to about $1\ \mathrm{W/m^2}$ at the pain threshold — twelve orders of magnitude. The decibel level $\beta$ compresses this into the formula below, where $I_0 = 10^{-12}\ \mathrm{W/m^2}$ is the reference intensity at the threshold of hearing.

Each factor of 10 in intensity is exactly $+10$ dB; doubling is $10\log_{10} 2 \approx +3$ dB. So 60 dB is a million times the intensity of 0 dB — the numbers look tame because the logarithm did the heavy lifting.

The Richter scale: earthquakes in logs

Charles Richter's magnitude is defined as $M = \log_{10}(A/A_0)$, the log of the measured wave amplitude against a reference. Each whole-number step multiplies the shaking amplitude by 10 — and because energy scales roughly as amplitude to the 3/2 power, one magnitude step releases about $10^{1.5} \approx 32$ times the energy.

A magnitude 7 quake is not 'a bit bigger' than a magnitude 5: it shakes 100 times harder and releases roughly $32 \times 32 \approx 1000$ times the energy. The log scale hides violence behind small numbers — always unpack it.

Natural logs and half-life

Exponential decay $N = N_0 e^{-\lambda t}$ surrenders to the natural log: take $\ln$ of both sides and you get $\ln(N/N_0) = -\lambda t$, a straight line with slope $-\lambda$. This is how labs actually measure decay constants — plot the log of the counts, read the slope.

The half-life drops out by setting $N/N_0 = 1/2$: $t_{1/2} = \ln 2 / \lambda$. Whenever you see $e$ and $\ln$ together in physics, a differential equation with proportional growth or decay is nearby.

pH and the habit of log thinking

Acidity is $\mathrm{pH} = -\log_{10}[\mathrm{H}^+]$, the negative log of hydrogen-ion concentration. Each pH unit is a factor of 10: lemon juice at pH 2 is a thousand times more acidic than pure water at pH 7. The minus sign just keeps everyday values positive.

Decibels, Richter, pH, stellar magnitudes, half-lives — the pattern repeats because human senses and natural phenomena both span multiplicative ranges. When quantities multiply, logs add, and addition is what brains handle gracefully.

Variables and units

Common mistakes

Real-world applications

• Audio engineers set gain, compression, and speaker specs entirely in decibels.

• Seismologists quote earthquake size on the logarithmic Richter and moment-magnitude scales.

• Chemists read acidity straight off the logarithmic pH scale.

• Nuclear physicists extract half-lives by plotting the natural log of count rates.