What is \[log 5\] ? How can we find logs of numbers without using a calculator?

Hint: Logs (or) logarithms are nothing but another way of expressing exponents. We will use the log tables to find the value of \[log 5\] , and for that we need to use the logarithm identities. Here we will use the division rule: log (a/b) = log a – log b. Then, find the values from the log table and calculate using a calculator we will get the final output. Next, we will show how to find the value of \[log5\] without using a calculator.

Complete step-by-step answer:
Simplest way is to calculate \[log5\] by referring to logarithmic tables, which shows \[\;log5 = 0.6990\].
Another way could be using \[\log 2 = 0.3010\] and \[\log 10 = 1\] (again for this we need log tables).
Suppose we need to find the value of \[log5\]:
Here, we will use the logarithm identity: \[log\left( {\dfrac{a}{b}} \right) = loga-logb\]
So, \[log5\] can be expressed as
\[log5 = log\left( {\dfrac{{10}}{2}} \right)\]
\[ = \log 10 - \log 2\]
Substituting the values of log, we will get,
\[ = 1 - 0.3010\]
\[ = 0.6990\]
Hence \[\log 5 = 0.6990\].
Next, we will find the value without using the calculators:
It is often needed to get the logarithm of a number without log tables.
So, it is good to know of the techniques which will be useful in finding the logarithm of many numbers.
Let us see how to find logarithmic values without the use of a calculator.
In a common logarithmic function, the base of the logarithmic function is 10. i.e. \[lo{g_{10}}\]or log represents this function. And in natural logarithmic function, the base of the logarithmic function is e. i.e. \[lo{g_e}\]or ln represents this function.
In fact one needs to remember log (to the base 10) for the first ten numbers and it makes things a lot easier.
\[log1 = 0\]
\[log10 = 1\]
Also, we also know that
\[log2 = 0.3010\]
\[log3 = 0.4771\]
\[log7 = 0.8451\]
\[loge = 0.693\]
Learn the above logarithms, as they will be useful in solving the logarithm of other numbers that are frequently required in various exams.
Then all logs can be worked out using these as
\[log4 = 2log2\]
\[log5 = 1 - log2\]
\[log6 = log2 + log3\]
\[log8 = 3log2\]
\[log9 = 2log3\]
In this way, we can find the other logarithmic values.
Using this technique, it is easy to find the logarithm of large numbers. But there are some numbers for which we cannot use this method. For example \[\log 11\].

Note: A logarithm is defined as the power to which number must be raised to get some other values. It is the most convenient way to express large numbers. There are three logarithm identities which one should know. They are:
1) Product rule: log (ab) = log a + log b
2) Quotient rule: log (a/b) = lag a – log b
3) Power rule: log (a^b) = bloga
Logarithms are also said to be the inverse process of exponential. The basic advantage of using logarithm base 10 is that they are easy to compute mentally for some special values. Natural logs are easier to use for theoretical work. They are easy to calculate numerically.