Log-log plots and the straight line approximation concept will come up again when we look at filters and amplifiers and frequency response. Note that the second line segment has slope -1, corresponding to x − 1 In the numerator or denominator) into straight line approximations that work over many orders of magnitude. That’s the power of log-log plotting: it turns rational polynomials (fractions with x n We’ve taken a complicated-looking fraction and turned it into a collection of straight line segments by using two tools: log-log plots and asymptotic approximations. In the middle, neither approximation is excellent, but the error can be quantified now that we know it’s there. That we used for computing various approximations in the Algebraic Approximations section. If plotting with a logarithmic scale on the y-axis is powerful, then plotting with a logarithmic scale on both the x-axis and y-axis is really powerful.
Now, while it’s not smooth or consistent or risk free, you’d see that in fact the general trend of growth (in a %/year growth sense) has been present for many decades. Now, click “Edit Graph”, click “Format”, and check the box next to “Log Scale” if the horizontal scale resets you may have to click “Max” again. The linear graph looks absolutely tiny for the first half of the dataset. To see for yourself, load this data, click “Max” to set the maximum timescale from 1970 onwards. This reflects the idea that in general, in any given year, the company’s profit (whether retained earnings or distributed as dividends) will be roughly a few percent of its current value. If we plot with a logarithmic scale, it reveals that in fact an investor at roughly any point has roughly doubled their investment value in about a decade. (It should also be plotted with dividends reinvested to show total returns, though this is a separate issue we can’t address here.) Why logarithmic? Because if we plot with a linear scale, it looks like all the growth has happened in just the last few years. The logarithm has transformed an exponential (with base 1.05) into a straight line.įor a similar example, plotting the value of a stock or mutual fund should be done on a semi-log plot. It turns out that these two examples are actually referring to the same thing: Saying “minus 6 dB per octave” might mean that the output signal at 200 Hz is only 1 2 For example 100 Hz and 200 Hz are one octave apart. Saying “minus 20 dB per decade” might mean that the output signal at 1000 Hz is only 1 10Īs large in voltage amplitude as the signal at 100 Hz.Īn octave is a 2X increase in frequency, just as it is in music.
In future sections when we talk about frequency response of amplifiers and filters, we will sometimes consider numbers like “-20 dB/decade” or “-6 dB/octave”.Ī decade is a 10X increase in frequency, for example from 100 Hz to 1000 Hz. , then “-60 dBV” is simply a short way of saying “1 millivolt.”īe careful to track whether a decibel unit is relative or absolute, and whether the absolute reference represents a power or voltage level. In another example, “dBV” means “decibels relative to 1 Volt.” Therefore, -60 dBv represents “-60dB relative to 1 volt.” As -60 dB in voltage is a voltage factor of 10 − 3
However, it is often convenient to define the denominator as a specific unit-bearing quantity.įor example, “dBm” means “decibels relative to 1 milliwatt.” Therefore, +20 dBm represents “+20dB relative to 1 milliwatt.” As Watts are a unit of power, and +20 dB is a multiplication of 100X in power, then “+20 dBm” is simply a short way of saying “100 milliwatts.” We can add the decibel values to get +40dB overall voltage gain – a factor of 10 2Īny unit specified only as “dB” refers only to a relative factor between signals. If we connected two amplifiers together in series, each of which had 10x voltage gain, then each would contribute +20dB gain. That’s why engineers speak colloquially about +3 dB or +6 dB to refer to a power gain of 2 or a voltage gain of 2, respectively.
These exact decibel values for multiplication by 2Īre commonly truncated.