- Let x= 2.3713018568 and y= 0.902780337. Find xy to three decimal places using
common logarithms. - Approximate the product xy from Prob. 1 using natural logarithms. Show that the result
is the same as that obtained with common logs when rounded off to three decimal
places. - The power gain of an electronic circuit, in units called decibels (abbreviated dB), can be
calculated according to this formula:
G= 10 log (Pout/Pin)where G is the gain, Pout is the output signal power, and Pin is the input signal power,
both specified in watts. Suppose the audio input to the left channel of high-fidelity
amplifier is 0.535 watts, and the output is 23.7 watts. What is the power gain of this
circuit in decibels? Round off the answer to the nearest tenth of a decibel.
- Suppose the audio output signal in the scenario of Prob. 3 is run through a long length
of speaker wire, so instead of the 23.7 watts that appears at the left-channel amplifier
output, the speaker only gets 19.3 watts. What is the power gain of the length of
speaker wire, in decibels? Round off the answer to three decimal places. - If a positive real number increases by a factor of exactly 10, how does its common (base-10)
logarithm change? - Show that the solution to Prob. 5 is valid for all positive real numbers.
- If a positive real number decreases by a factor of exactly 100 (becomes 1/100 as great),
how does its common logarithm change? - Show that the solution to Prob. 7 is valid for all positive real numbers.
- If a positive real number is divided by a factor of 357, how does its natural (base-e)
logarithm change? Express the answer to two decimal places. - Show that the solution to Prob. 9 is valid for all positive real numbers.
Practice Exercises 497