When quantities range from tiny to huge, scientists measure them on a scale of exponents, a logarithmic scale. You already read log graphs; now you read log scales.
Key idea
On a log scale, each step of 1 means multiplying by the base (usually 10). pH: $\text{pH} = -\log[H^+]$, so one pH unit is a factor of 10 in acidity. Decibels: $L = 10\log\frac{I}{I_0}$, so 10 dB is a factor of 10 in intensity. A difference of $d$ on the scale means a ratio of $10^d$ (for decibels, $10^{d/10}$).
Why it works
The log of a product is a sum, so multiplying a quantity by 10 adds 1 to its log. That turns a range like $10^{-14}$ to $1$ into a tidy scale from 14 to 0. The price is that the scale does not add up the way it looks: going from pH 4 to pH 2 is not "twice as acidic," it is $10^2 = 100$ times.
Worked examples
Worked example: Acidity
How many times more acidic is pH 4 than pH 8?
- $[H^+] = 10^{-4}$ versus $10^{-8}$.
- Ratio: $\frac{10^{-4}}{10^{-8}} = 10^4$.
Answer: 10,000 times
Worked example: Loudness
A sound is 30 dB louder than another. How many times greater is its intensity?
- $10\log\frac{I_1}{I_2} = 30$, so $\log\frac{I_1}{I_2} = 3$.
- $\frac{I_1}{I_2} = 10^3$.
Answer: 1,000 times
Common mistake
Reading a log scale as if it were linear. A pH of 4 is not "twice as acidic" as a pH of 8. The difference is 4 units, and each unit is a factor of 10, so it is $10^4 = 10{,}000$ times as acidic. Subtract the scale values, then raise 10 to that power.