Converting ratios into decibels

Statement

Convert the intensity ratio $P/P_0$ into dB for: (a) $P=P_0$; (b) $P = 2P_0$; (c) $P = 0{,}5\,P_0$; (d) $P = 10^4 P_0$.

Solution

Apply $L = 10\log_{10}(P/P_0)$:

• (a) $L = 10\log_{10} 1 = \boxed{0\ \text{dB}}$;
• (b) $L = 10\log_{10} 2 \approx \boxed{3\ \text{dB}}$;
• (c) $L = 10\log_{10} 0{,}5 \approx \boxed{-3\ \text{dB}}$;
• (d) $L = 10\log_{10} 10^4 = 10\cdot 4 = \boxed{40\ \text{dB}}$.

Note the symmetry between (b) and (c): doubling adds about $3$ dB, halving removes about $3$.

Links

Topics: Logarithmic function
Concepts: Decibel · Logarithm · Logarithmic scale
Functions: Logarithmic function
Methods: Decibel
Skills: Calculating
Exercise type: Evaluating expressions