Logarithmic scale — decibels, Richter, pH: Question bank

True or false — justify

Each statement is either true or false. Say which, then give the one-sentence reason — the reason is the point, not the verdict.

A magnitude 8 earthquake shakes the ground twice as hard as a magnitude 4

False. The scale is logarithmic, so the amplitude ratio is

1 0^{8 - 4} = 1 0^{4} = 10, 000

times, not

8/4 = 2

Two identical 50 dB speakers together produce 100 dB

False. Decibels are logarithms and don't add; doubling the intensity adds only

10 lo g_{10} 2 \approx 3

dB, giving 53 dB.

A pH of 0 is impossible because you can't take the log of a negative number

False. pH 0 means

[H^{+}] = 1 0^{0} = 1

mol/L, a perfectly valid (very acidic) concentration; the log's argument is positive.

Moving from pH 3 to pH 5 makes a solution 100 times less acidic

True.

[H^{+}]

falls by

1 0^{5 - 3} = 100

, and less

H^{+}

means less acidic — the minus sign flips the direction.

On any log scale, equal steps correspond to equal ratios of the underlying quantity

True. Because

Δ L = k lo g_{10} r

depends only on the multiplier

r

, one fixed step always means one fixed ratio, wherever you start.

0

dB means "total silence"

False.

0

dB means

I = I_{0}

, the threshold of hearing (

10 lo g_{10} 1 = 0

); true silence would be

I = 0

, whose log is

- \infty

Halving sound intensity changes the level by

- 3

True.

Δ L = 10 lo g_{10} (1/2) = - 10 lo g_{10} 2 \approx - 3.01

dB — the exact mirror of the "+3 dB doubling" rule.

The Richter scale can never produce a negative magnitude

False. If ground amplitude

A < A_{0}

then

lo g_{10} (A / A_{0}) < 0

, giving a genuine negative magnitude for very tiny tremors.

Decibels have physical units of watts

False. A decibel is a pure number (a ratio inside a log); the physical units of intensity cancel in

I / I_{0}

, so dB is dimensionless.

Spot the error

Each line contains a flawed step. Find where the reasoning breaks and state the fix.

"pH

= - lo g_{10} (4 \times 1 0^{- 3}) = - lo g_{10} 4 \times lo g_{10} 1 0^{- 3}

The product rule gives a sum, not a product:

lo g (ab) = lo g a + lo g b

, so it should be

- (lo g_{10} 4 + lo g_{10} 1 0^{- 3})

"60 dB source plus a 40 dB source = 100 dB total"

You cannot add dB; convert to intensities

I_{0} 1 0^{6}

and

I_{0} 1 0^{4}

, add those to get

I_{0} 1 0^{6} (1.01)

, then

10 lo g_{10}

gives ≈60.04 dB — the quiet source barely matters.

"Magnitude ratio

M_{7} / M_{5} = 7/5 = 1.4

, so a 7 is 1.4× a 5"

You must subtract then exponentiate:

A_{7} / A_{5} = 1 0^{7 - 5} = 100

; ratios of scale values are meaningless, ratios of the quantity are what count.

"To un-log

L = 10 lo g_{10} (I / I_{0})

, write

I = I_{0} 1 0^{L}

The factor of 10 was divided off first:

L /10 = lo g_{10} (I / I_{0})

, so

I = I_{0} 1 0^{L /10}

, not

1 0^{L}

"pH 7 water has zero hydrogen ions because

lo g

gave 7, a whole number"

pH 7 means

[H^{+}] = 1 0^{- 7}

mol/L — small but nonzero; a whole-number pH just means the concentration is an exact power of ten.

"Since

lo g (a + b) = lo g a + lo g b

, two 60 dB sources give

10 lo g_{10} (1 0^{6} + 1 0^{6}) =

... wait that's fine"

The identity quoted is wrong (

lo g (a + b) \neq = lo g a + lo g b

), but the method is right — you correctly add the intensities

1 0^{6} + 1 0^{6}

inside the log; just never claim the false identity.

"Each pH unit is ×10 acidity, so pH 2 is

2 \times 10 = 20

times pH 1's acidity"

Each unit is ×10 per step, and there is one step from 1 to 2, so it's

1 0^{2 - 1} = 10

times, not

20

; you multiplied instead of using the exponent.

Why questions

Explain the reason, not just the rule.

Why does the change

Δ L

depend only on the ratio

r

and not on the starting intensity?

Because the product rule splits

lo g (r I / I_{0}) = lo g r + lo g (I / I_{0})

, so the starting term cancels in the subtraction, leaving

Δ L = k lo g_{10} r

Why is there a minus sign in pH but not in decibels?

[H^{+}]

is a tiny number like

1 0^{- 3}

whose log is negative; the minus flips it to a friendly positive value so "more acid → lower pH" reads naturally.

Why choose base 10 for all three scales rather than base

e

Base 10 makes "one step = one power of ten" line up with our decimal intuition about orders of magnitude; any base works but 10 reads cleanest.

Why did engineers use deci-bels instead of bels?

One whole bel (a factor of ×10) is too coarse for fine loudness distinctions, so tenths of a bel — decibels — give a usable resolution, which is where the factor of 10 comes from.

Why do our ears, earthquakes, and acids all suit a log scale?

They respond multiplicatively — equal ratios feel like equal steps (the Weber–Fechner idea) — and a log is exactly the tool that turns equal ratios into equal additive steps.

Why can't you just add two speakers' decibel readings?

dB is already a logarithm of intensity; only the physical intensities are additive, and

lo g (I_{1} + I_{2}) \neq = lo g I_{1} + lo g I_{2}

, so you must add intensities first.

Why is +3 dB such a famous "round" number to audio people?

Because

10 lo g_{10} 2 \approx 3.01

, so +3 dB is almost exactly a doubling of intensity — a memorable shortcut straight from the product rule.

Edge cases

Boundary and degenerate inputs — where naive rules wobble.

What is the dB level of perfect silence,

I = 0

Undefined /

- \infty

lo g_{10} (0)

has no value, so the scale simply has no bottom — the "0 dB" floor is the threshold of hearing, not silence.

What Richter magnitude does a wave with

A = A_{0}

give?

Exactly

0

, since

lo g_{10} (A_{0} / A_{0}) = lo g_{10} 1 = 0

— the reference amplitude is the natural zero of the scale.

Can pH exceed 14 or drop below 0?

Yes — 0–14 is just the common range for dilute solutions; concentrated acids can give negative pH and strong bases can exceed 14 because

[H^{+}]

can be

> 1

or extremely small.

What happens to

Δ L

if the quantity stays the same,

r = 1

Δ L = k lo g_{10} 1 = 0

— no change in level, exactly as expected when nothing multiplies.

If intensity is divided by 10, what is

Δ L

in dB?

r = 1/10

gives

Δ L = 10 lo g_{10} (0.1) = - 10

dB — the scale drops by a full ten, mirroring the +10 dB of a ×10 increase.

A solution's

[H^{+}]

is halved — how does pH change?

pH rises by

- lo g_{10} (1/2) = + 0.30

; halving the acid nudges pH up by about 0.3, the pH twin of the "3 dB" fact.

What magnitude corresponds to zero ground shaking,

A = 0

Undefined (

lo g_{10} 0 = - \infty

); a completely still ground has no magnitude, so the scale only describes actual (nonzero) motion.

Two sources differ by 20 dB — how do their intensities compare, and does the quiet one matter when combined?

1 0^{20/10} = 100 \times

apart; adding the quiet one changes the total by

10 lo g_{10} (1.01) \approx 0.04

dB, i.e. negligibly.

Connections

Parent — the topic these traps drill.
Laws of Logarithms — the product/quotient rules behind nearly every fix above.
Exponential Functions — the inverse used when we un-log a scale value.
Change of Base Formula — why base 10 is the natural choice.
Weber–Fechner Law — the perceptual "why" for decibels.
Orders of Magnitude — "counting powers of ten" is the log scale in one phrase.