We can either compress a group of logs into a single log or expand a single log into a group of logs using the above rules of logs. Also, to convert a negative log into a positive log, we can take the reciprocal of the base, i.e., I.e., To convert a negative log into a positive log, we can just take the reciprocal of the argument. We can calculate this using the power rule of logarithms. The negative logs are of the form −log b a. When a number is raised to log whose base is same as the number, then the result is just the argument of the logarithm. It is a kind of canceling log from both sides. This rule is used while solving the equations involving logarithms. Then we get: log b a = (log a) / (log b). Hence we can change the base to 10 as well. Using this property, we can change the base to any other number. It says:Īnother way of writing this rule is log b a The base of a logarithm can be changed using this property. This resembles/is derived from the power of power rule of exponents: (x m) n = x mn. Here, the bases must be the same on both sides. The exponent of the argument of a logarithm can be brought in front of the logarithm, i.e., This resembles/is derived from the quotient rule of exponents: x m / x n = x m-n. Note that the bases of all logs must be the same here as well. The logarithm of a quotient of two numbers is the difference between the logarithms of the individual numbers, i.e., This resembles/is derived from the product rule of exponents: x m ⋅ x n = x m+n. Note that the bases of all logs must be the same here. The logarithm of a product of two numbers is the sum of the logarithms of the individual numbers, i.e., Thus, the logarithm of any number to the same base is always 1. Since a 1 = a, for any 'a', converting this equation into log form, log a a = 1. When we extend this to the natural logarithm, we have, since e 0 = 1 ⇒ ln 1 = 0. Obviously, when a = 10, log 10 1 = 0 (or) simply log 1 = 0. Converting this into log form, log a 1 = 0, for any 'a'. Because from the properties of exponents, we know that, a 0 = 1, for any 'a'. The value of log 1 irrespective of the base is 0. Let us see each of these rules one by one here. If you want to see how all these rules are derived, click here. Here are the rules (or) properties of logs. The rules of logs are used to simplify a logarithm, expand a logarithm, or compress a group of logarithms into a single logarithm. Observe that we have not written 10 as the base in these examples, because that's obvious. In other words, it is a common logarithm. I.e., if there is no base for a log it means that its log 10. But usually, writing "log" is sufficient instead of writing log 10. i.e.,Ĭommon logarithm is nothing but log with base 10. But it is not usually represented as log e. Natural logarithm is nothing but log with base e. These two logs have specific importance and specific names in logarithms. Observe the last two rows of the above table. Here is a table to understand the conversions from one form to the other form. This is called " log to exponential form" This is called " exponential to log form" The above equation has two things to understand (from the symbol ⇔):
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