714 Worked-Out Solutions to Exercises: Chapters 21 to 29
- We can use the property of natural logarithms that converts a product into a sum:
lnxy= ln x+ ln y
In this case, x= 2.3713018568 and y= 0.902780337. Therefore:
ln (2.3713018568 × 0.902780337)
= ln 2.3713018568 + ln 0.902780337
≈ 0.86343911100 + (−0.102276014)
≈ 0.86343911100 − 0.102276014
≈ 0.761163097This is the natural logarithm of the product we wish to find. If we find the natural
antilogarithm of this, we’ll get the desired result. Inputting this to a calculator and then
rounding to three decimal places:antiln (0.761163097) ≈ 2.141- In this situation, Pout= 23.7 and Pin= 0.535. We can plug these numbers into the
formula for gain G in decibels (dB), and then round off as follows:
G= 10 log 10 (Pout/Pin)
= 10 log 10 (23.7 / 0.535)
= 10 log 10 44.299
≈ 10 × 1.6464
≈ 16.5 dB- In this situation, Pout= 19.3 and Pin= 23.7. We’re interested in the power gain of the
speaker wire, not the power gain of the amplifier. We can plug these numbers into the
formula for gain G in decibels, and then round off as follows:
G= 10 log 10 (Pout/Pin)
= 10 log 10 (19.3 / 23.7)
= 10 log 10 (0.81435)
≈ 10 × (−0.089189)
≈ −0.892 dB- If a positive real number increases by a factor of exactly 10, then its common logarithm
increases (it becomes more positive or less negative) by exactly 1. - Let x be the original number, and let y be the final number. We’re told that y= 10 x.
Taking the common logarithm of each side of this equation gives us
logy= log 10 (10x)