Logarithmic Functions and Their Graphs:
Inverse of Exponential Functions:
We stated in the part on exponential functions, which exponential functions were one-to-one. One-to-one functions had the unique property which they have inverses that are as well functions. And, as most of you said in class, one-to-one functions can applied to both the sides of an equation. They as well pass a horizontal line test.
This part is about the inverse of exponential function. The inverse of an exponential function is the logarithmic function. Keep in mind that the inverse of a function is received by switching x and y coordinates. It reflects the graph regarding the line y = x. As you can state from the graph to the right, the logarithmic curve is a reflection of exponential curve.
The table below illustrates how x and y values of points on an exponential curve can be switched to determine the coordinates of the points on logarithmic curve.
Comparison of Exponential and Logarithmic Functions:
Let us look at some of the properties of two functions.
The standard form for logarithmic function is: y = loga x
Note, if the ‘a’ in expression above is not a subscript (that is, lower than the ‘log’), then you require to update your web browser.
Working Definition of Logarithm:
In an exponential function, the x was exponent. The main aim of inverse of a function is to state you what x value were used when you already know the y value. Therefore, the aim of the logarithm is to state you the exponent.
Therefore, our simple definition of a logarithm is that it is the exponent.
The other way of looking at expression ‘loga x’ is ‘to what power (or exponent) should a be increased to get x?’
The logarithmic form of equation y = loga x is equivalent to the exponential form x = ay.To rewrite one form in other, keep the base similar, and switch sides with other two values.
Properties of Logarithms:
loga 1 = 0 since a0 = 1
No matter what the base is, as long as it is legal, the log of 1 is for all time 0. That is as logarithmic curves always pass via (1,0)
loga a = 1 since a1 = aAny of the value increased to the first power is that similar value.
loga ax = x
The log base a of x and a to x power are inverse functions. When inverse functions are applied to one other, they inverse out, and you are left with argument, in this case, x.
loga x = loga y implies that x = y
When two logs with similar base are equivalent, then the arguments should be equal. loga x = logb x implies that a = b
When two logarithms with similar argument are equivalent, then the bases should be equal.
Common Logs and Natural Logs:
There are two logarithm buttons on your calculator. One is marked as ‘log’ and the other is marked as ‘ln’. Neither one of such consists of the base written in. The base can be recognized, though, by looking at inverse function that is written above the key and accessed by the 2nd key.
Common Logarithm (base 10):
Whenever you see ‘log’ written, with no base, suppose the base is 10.
That is: log x = log10 x.Some of the applications which use common logarithms are in pH (to compute acidity), decibels (for sound intensity), the Richter scale (for earthquakes).
The interesting (possibly) side note regarding pH. Sewers of the Village of Forsyth Code require forbids the release of waste with a pH of less than 5.5 or higher than 10.5
The common logs as well serve the other aim. Every rise of 1 in a common logarithm is the outcome of 10 times the argument. That is, an earthquake of 6.3 has 10-times the magnitude of 5.3 earthquakes. The decibel level of loud rock music or the chainsaw (115 decibels = 11.5 bels) is 10-times louder than the chickens within a building (105 decibels = 10.5 bels).Natural Logarithms (base e):
Keep in mind that number e that we had from the prior part? You know the one that was around 2.718281828 (however does not repeat or terminate). That is the base for natural logarithm.
ln x = loge x
The exponential growth and decay models are one of the applications which use natural logarithms. This comprises continuous compounding, radioactive decay (that is, half-life), and population growth. Usually applications are procedure which is continually happening. Now, such applications were first illustrated in the exponential part, however you will be capable to solve for other variables included by using logarithms.
In higher level of mathematics, the natural logarithm is a logarithm of choice. There are some special properties of natural logarithm function, and its inverse function, that make life much simpler in calculus.
As ‘ln x’ and ‘ex’ are inverse functions of one other, any time an ‘ln’ and ‘e’ appear right next to one other, with absolutely nothing in between them (that is, whenever they are composed with one other), then they inverse out, and you are left with argument.
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