Logarithms And Antilogarithm
If log M = x, then M is called the antilogarithm of x and is written as M = antilog x.
For example, if log 39.2 = 1.5933, then antilog 1.5933 = 39.2.
If the logarithmic value of a number be given then the number can be determined from the antilog-table. Antilog-table is similar to log-table; only difference is in the extreme left-hand column which ranges from .00 to .99.
Example on antilogarithm:
1. Find antilog 2.5463.
Solution:
Vividly, we are to find the number whose logarithm is 2.5463. For this consider the mantissa .5463. Now find .54 in the extreme left-hand column of the antilog-table. You would obtain this from four-figure antilog-table. Go horizontally to the right to the column headed by 6 of the top-most row and read the number 3516. Again, we move along the same horizontal line further right to the column headed by 3 of mean difference and read the number 2 there. This 2 is currently added to the earlier number 3516 to give 3518. Due to the fact that the characteristic is 2, there supposed to be three digits in the integral part of the needed number.
Thus, antilog 2.5463 = 351.8.
2. If log x = -2.0258, find x.
Solution:
In order to find the value of x using antilog-table, the decimal part (i.e., the mantissa) must be made positive. For this we move ahead as follows:
log x = -2.0258 = - 3 + 3 - 2.0258
= - 3 + .9742 =3.9742
Thus, x = antilog 3.9742.
Now, from antilog table, we obtain the number corresponding to the mantissa
.9742 as (9419 + 4) = 9423.
Again the characteristic in log x is (- 3).
Thus, there ought to be two zeroes between the decimal point and the first noteworthy digit in the value of x.
Therefore, x = .009423.
The idea of logarithms is to reverse the operation of exponential function, that is, raising a number to a power. For instance, the third power (or cube) of 2 is 8, due to the fact that 8 is the product of three factors of 2:
It follows that the logarithm of 8 with regard to base 2 is 3, so log2 8 = 3.
Exponentiation
The third power of a few number b is the product of three factors of b. Generally, raising b to the n-th power, where n is a natural number, is carried out by multiplying n factors of b. The n-th power of b is written bn. Exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For instance, b−1 is the reciprocal of b, that is, 1/b. For more details, including the formula bm + n = bm • bn, see exponential or for an elementary treatise.
The logarithm of a positive real number x with regard to base b, a positive real number not equal to 1, is the exponent by which b ought to be raised to yield x. On the contrary, the logarithm of x to base b is the solution y to the equation
The logarithm is denoted "logb(x)" (pronounced as "the logarithm of x to base b" or "the base-b logarithm of x"). In the equation y = logb(x), the value y is the answer to the question "to what power must b be raised, in order to yield x. This question can as well be addressed with a richer answer for composite numbers, which is done in section "Complex logarithm", and this answer is much more expansively investigated in the page for the multifaceted logarithm.
Examples
For instance, log2(16) = 4, since 24 = 2 ×2 × 2 × 2 = 16. Logarithms can as well be negative. A third example: log10(150) is just about 2.176, which lies between 2 and 3, just like150 lies between 102 = 100 and 103= 1000. At the end, for any base b, logb(b) = 1 and logb(1) = 0, since b1 = b and b0 = 1, correspondingly.
Logarithmic identities
A number of significant formulas, sometimes known as logarithmic identities or log laws, relate logarithms to one another.
Product, quotient, power and root
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples. Every one of the identities can be derived after substitution of the logarithm definitions x=blogb(x) or y=blogb(y)in the left hand sides.
| Formula | Example | |
| product | log3(243)=log3(9.27)=log3(9)+log3(27) | |
| quotient | logb(x/y)=logb(x)-logb(y) | log2(16)=log2(64/4)= log2(64)- log2(4)=6-2=4 |
| power | ||
| root |
Change of base
The logarithm logb(x) can be calculated from the logarithms of x and b with respect to an arbitrary base k with the use of the formula below. Typical scientific calculators calculate the logarithms to bases 10 and e. Logarithms with regard to any base b can be determined with the use of these two logarithms by the earlier formula. Given a number x and its logarithm logb(x) to an unknown base b, the base is given by:
Special bases
Among all choices for the base, three are especially common. These are b = 10, b = e (the irrational mathematical constant ≈ 2.71828), and b = 2. In mathematical analysis, the logarithm to base e is widespread due to its specific analytical properties discussed below. On the contrary, base-10 logarithms are easy to use for manual calculations in the decimal number system. Therefore, log10(x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly larger than log10(x). For instance, log10(1430) is roughly 3.15.
The next integer is 4, which is the number of digits of 1430. Both the natural logarithm and the logarithm to base two are used in information theory, corresponding to the use of nats or bits as the basic units of information, correspondingly. Binary logarithms are as well used in computer science, where the binary system is everywhere, in music theory, where a pitch ratio of two (the octave) is everywhere and the cent is the binary logarithm (scaled by 1200) of the ratio between two adjacent equally-tempered pitches, and in photography to measure exposure values.
The table below lists common notations for logarithms to these bases and the fields where they are used. A lot of disciplines write log(x) rather than logb(x), when the intended base can be found from the context. The notation blog(x) as well occurs. The "ISO notation" column lists designations recommended by the International Organization for Standardization (ISO 31-11).
| Base b | Name for logb(x) | ISO notation | Other notations | Used in |
| 2 | binary logarithm | lb(x) | ld(x), log(x), lg(x), log2(x) | computer science, information theory, music theory, photography |
| e | natural logarithm | ln(x) | log(x) (in mathematics and a lot of programming languages) | mathematics, physics, chemistry, statistics, economics, information theory, and a few engineering fields |
| 10 | common logarithm | lg(x) | log(x), log10(x) (in engineering, biology, astronomy) | A lot of engineering fields like decibel, logarithm tables, handheld calculators, spectroscopy |
Logarithmic function
To substantiate the definition of logarithms, it is essential to show that the equation
bx=y
has a solution x and that this solution is inimitable, in so far as y is positive and b is positive and unequal to 1. A proof of that fact needs the intermediate value theorem from elementary calculus. This theorem states that a continuous function that produces two values m and n as well yields any value that lies between m and n. A function is continuous if it does not "jump", that is, if its graph can be drawn without lifting the pen.
This property can be seen to hold for the function f(x) = bx. Due to the fact that f takes illogically large and illogically small positive values, any number y > 0 lies between f(x0) and f(x1) for appropriate x0and x1. Therefore, the intermediate value theorem sees to it that the equation f(x) = y has a solution. Moreover, there is just one solution to this equation, due to the fact that the function f is severely increasing (for b > 1), or firmly decreasing (for 0 < b < 1).
The exclusive solution x is the logarithm of y to base b, logb(y). The function that assigns to y its logarithm is known as logarithm function or logarithmic function (or mere logarithm).
The function logb(x) is basically characterized by the above product formula
logb(xy)=logb(x)+logb(y)
More correctly, the logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f(b) = 1 and
Inverse function
The graph of the logarithm function logb(x) is obtained by reflecting the graph of the function bx (red) at the diagonal line (x = y).
The formula for the logarithm of a power particularly says that for any number x,
logb(bx)=xlogb(b)=x
In prose, taking the x-th power of b and then the base-b logarithm produces back x. On the other hand, given a positive number y, the formula
b logb(y)=y
indicates that first taking the logarithm and then converting to exponent gives back y. Therefore, the two possible ways of combining (or composing) logarithms and exponent produces back the original number. Thus, the logarithm to base b is the inverse function of f(x) = bx.
Inverse functions are strictly related to the original functions. Their graphs correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as illustrated at the right: a point (t, u = bt) on the graph of f yields a point (u, t = logbu) on the graph of the logarithm and vice versa. As a result, logb(x) diverges to infinity (gets bigger than any specific number) if x grows to infinity, in so far as b is greater than one. In that case, logb(x) is an increasing function. For b < 1, logb(x) tends to rather minus infinity. When x approaches zero, logb(x) moves to minus infinity for b > 1 (plus infinity for b < 1, correspondingly).
Derivative and antiderivative
The graph of the natural logarithm –the green curve and its tangent at x = 1.5 –the black curve.
Analytic properties of functions pass to their inverses. Therefore, as f(x) = bx is a continuous and differentiable function, therefore is logb(y). Approximately, a nonstop function is differentiable if its graph has no sharp "corners". Furthermore, as the derivative of f(x) evaluates to ln(b)bx by the properties of the exponential function, the chain rule signifies that the derivative of logb(x) is given by
That is, the slope of the tangent touching the graph of the base-b logarithm at the point (x, logb(x)) equals 1/(x ln(b)). Particularly, the derivative of ln(x) is 1/x, which implies that the antiderivative of 1/x is ln(x) + C. The derivative with a generalized functional argument f(x) is
The quotient at the right hand side is known as the logarithmic derivative of f. Computing f'(x) by means of the derivative of ln(f(x)) is referred to as logarithmic differentiation. The antiderivative of the natural logarithm ln(x) is:
Equivalent formulas, such as antiderivatives of logarithms to other bases can be derived from this equation with the use of the change of bases.
Integral representation of the natural logarithm
The natural logarithm of t is the shaded area below the graph of the function f(x) = 1/x (reciprocal of x).
The natural logarithm of t agrees with the integral of 1/x dx from 1 to t:
Put another way, ln(t) equals the area between the x axis and the graph of the function 1/x, and ranges from x = 1 to x = t (figure at the right). This is a result of the basic theorem of calculus and the fact that derivative of ln(x) is 1/x. The right hand side of this equation can function as a definition of the natural logarithm. Product and power logarithm formulas can be derived from this definition. For instance, the product formula ln(tu) = ln(t) + ln(u) is construed as:
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