y = a ln (x) + b,where x and y are variables and a and b are constants. An example is:
y = 2.79 ln (x) + 5.80.This function can be expressed in many equivalent forms using the change of base formula. For example:
y = 2.79 ln (x) + 5.80
= 6.43 log (x) + 5.80
= 5 log 6 (8 x).
wavelength (in meters) name of region of EM spectrum 10 −15 10 −14 10 −13 gamma 10 −12 10 −11 10 −10 x-ray 10 −9 10 −8 10 −7 ultraviolet 10 −6 visible region 10 −5 infrared 10 −4 10 −3 10 −2 microwave 10 −1 10 0 short radio waves 10 1 FM radio 10 2 AM radio 10 3 10 4 10 5 long radio waves 10 6
N is called the gain of the amplifier or attenuator. It is measured in units called decibels (abbreviated db). Notice that the gain is positive for amplifiers, negative for attenuators and zero if Pout /Pin = 1.
Now put a factor of Pint /Pint inside the brackets:
Now use property 1 of logarithms:
The first term in the final expression is just the gain of amplifier # 1 and the second term is the gain of amplifier # 2. This proves that the gain of a system equals the sum of the gains of its components.
45 = 10 log 10 (Pout / 0.05W)Solving for Pout gives:
Pout = 0.05 W · 10 4.5 = 1581 W
where P is the pressure fluctuation of the sound and Po is the pressure fluctuation of a sound at the threshold of human hearing, which is 20 μPa.
We can eliminate Po and compare Ptraffic and Proom directly by subtracting these equations and using property 2 of logarithms:
Solving for the ratio Ptraffic /Proom gives:
The threshold intensity for human hearing is Io = 10 −12 Watts / m2. A factor of 10 appears in this formula (rather than 20) since the energy of a wave is proportional to the square of the pressure fluctuation.
where E is the energy released by the earthquake and Eo is a reference arbitrarily set at the limit of sensitivity of Richter's original seismic measuring apparatus.
where [ H + ] is the concentration in moles/liter of the H + ion responsible for acidity. A solution with pH = 7 is neutral, pH < 7 is an acid and pH > 7 is a base.
where I is the intensity of the light from the star. I1 is the intensity of a first magnitude star (the 20 brightest stars in the sky are approximately magnitude 1). The most powerful telescopes can detect stars as faint as magnitude +24.