The sound intensity of a soft whisper is about 1/200000 of the sound intensity of a shout. What is the decibel level of a whisper if a shout has a loudness level of about 85 dB?

L = 10log(i1/10^-12)

To find the decibel level of a soft whisper, we can use the formula for decibel level:

L = 10 * log10(i1 / i0)

Where L is the decibel level, i1 is the sound intensity of the soft whisper, and i0 is a reference sound intensity of 10^-12 watts/m^2.

In this case, we know that the sound intensity of a soft whisper is 1/200000 (or 1/2 x 10^-5) of the sound intensity of a shout. So we can substitute these values into the formula:

L = 10 * log10((1/2 x 10^-5) / 10^-12)

First, let's simplify the fraction:

L = 10 * log10((1/2) * (10^-5 / 10^-12))

Now, let's simplify the powers of 10 in the denominator:

L = 10 * log10((1/2) * (10^12 / 1))

The term (10^12 / 1) is equal to 10^12, so we can continue simplifying:

L = 10 * log10(1/2) + 10 * log10(10^12)

Now, let's calculate the logarithm values:

L = 10 * (-0.3010) + 10 * 12

The value of log(1/2) is approximately -0.3010, so we substitute that value:

L = -3.010 + 120

Now, we can calculate the sum:

L = 116.990 dB

Therefore, the decibel level of a soft whisper is approximately 116.990 dB.