# 9th Class Mathematics Chapter No 03 Exercise No 3.4 Question No 01

## Description

The logarithm is the inverse operation to exponentiation in mathematics. That means the logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. In simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000 (1000 = 10 × 10 × 10 = 103); 10 is used as a factor three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers b and x where b is not equal to 1. The logarithm of x to base b, denoted logb(x), is the unique real number y such that

by = x.

For example, as 64 = 26, then:

log2(64) = 6

The logarithm to base 10 (that is b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the number e (≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler derivative. The binary logarithm uses base 2 (that is b = 2) and is commonly used in computer science.

Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition because of the fact — important in its own right — that the logarithm of a product is the sum of the logarithms of the factors:

provided that b, x and y are all positive and b ≠ 1. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.

Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.

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. Each of the identities can be derived after substitution of the logarithm definitions x=b^{\log _{b}(x)}} x=b^{\log _{b}(x)} or y=b^{\log _{b}(y)}} y=b^{\log _{b}(y)} in the left hand sides.

product log{b}(xy)=\log _{b}(x)+\log _{b}(y)}

quotient log _{b}\!\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}

Change of base

The logarithm logb(x) can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:

log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.\,} \log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.\,

Typical scientific calculators calculate the logarithms to bases 10 and e.[4] Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:

log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.\,} \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.\,

Given a number x and its logarithm logb(x) to an unknown base b, the base is given by:

b=x^{\frac {1}{\log _{b}(x)}}.} b=x^{\frac {1}{\log _{b}(x)}}.

Particular bases

Among all choices for the base, three are particularly 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 because of its particular analytical properties explained below. On the other hand, base-10 logarithms are easy to use for manual calculations in the decimal number system:[5]

log _{10}(10x)=\log _{10}(10)+\log _{10}(x)=1+\log _{10}(x).\ } \log _{10}(10x)=\log _{10}(10)+\log _{10}(x)=1+\log _{10}(x).\

Thus, log10(x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly bigger than log10(x).[6] For example, log10(1430) is approximately 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.