SME
Grade 11 Science · Problem type
Sound level in decibels
Use this for “A busy street has a sound intensity of 10⁻⁵ W/m². What is its sound level in decibels?”
Diagram
0.00001 = 1 × 10⁻⁵, so its log is −5 plus the log of 1.
log₁₀ 0.00001
log₁₀ 0.00001 = −5 + log₁₀ 1 ≈ −5 + 0.000 = −5
The power of ten gives the whole part; the ruler gives the decimal part.
Sound level: 70 dB
This is an example. Type your own values to start a new problem.
β (dB)
=
10
log
I (W/m²)
10⁻¹²
W/m²
Dividing by 10⁻¹² adds 12 to the log: 10 log(I ÷ 10⁻¹²) = 10 × (log I + 12), the form the Formulas section works with L = log I.
L
Log of the intensity
worked out (not typed)
Formulas
L
=
log₁₀(
I
)
−5
=
log₁₀(1
×
10
-5
)
β
=
10
×
(
L
+
12)
70 = 10 × (−5 + 12)
Assumptions
•
Hearing starts at I₀ = 10⁻¹² W/m² (0 dB); β = 10 log(I ÷ I₀).
•
Each power of ten in intensity adds 10 dB: ten times the intensity, 10 dB louder.
Step-by-step
WE KNOW
I = 1 × 10⁻⁵ W/m²
FIND
log of the intensity (L), sound level (β)
Step 1 · Find log of the intensity (L)
Use L = log₁₀(I)
The power of ten that makes the intensity.
L
=
log₁₀(1
×
10
-5
)
L
=
−5
Step 2 · Find sound level (β)
Use β = 10 × (L + 12)
log(
I
10
-12
)
is
log
I
+
12:
the
powers
of
ten
above
the
threshold,
10
dB
each.
β
=
10
×
(−5
+
12)
β
=
10
×
7
β
=
70
dB
Check
Put the numbers back in. Both sides should match.
−5
=
log₁₀(1
×
10
-5
)
✓
70 = 10 × (−5 + 12) ✓
Topic check
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