Question

Solve “log(base10)5” without using a calculator?

Answer 1

Hint : Log is an identity in math which has its own expansion to find the values, such expansion is too lengthy that is why certain values of “log” like for “2” and “3”. For breaking large numbers into small, ”log” has properties which can be used to solve the questions.
Formulae Used: $\log (\dfrac{a}{b}) = \log a - \log b$ , $\log 10 = 1,\,\log 2 = 0.3010\,for\,base\,10$

Complete step-by-step answer :
To solve this question we need to use certain “log” formulas in which additive property of “log” can be seen, the described formulae is:
$\log (\dfrac{a}{b}) = \log a - \log b$
Vice versa of the above formulae is also true.
Now here for this question we are going to write “5” as in the ratio of “10 and 2” and after splitting the term the the modified question can be written as:
$\log (5) = \log (\dfrac{{10}}{2})$
On solving we get:

$$\log (5) = \log (\dfrac{{10}}{2}) = \log 10 - \log 2 \\\ = 1 - 0.3010(\log 10 = 1,\,\log 2 = 0.3010\,for\,base\,10) \\\ = 0.6990 \;$$

Our required answer is $0.6990$ , this value can be verified by seeing the “log” table.
So, the correct answer is “ $0.6990$ ”.

Note : Graph of “log” can also be drawn and see the value but its quite complicated method to obtain the values because you can’t draw proper “log” graph for any values you want except for “1” because value of “log1” is zero.
“Log” function works on its base given, so for the different values of “log” at different bases a ” log” table is defined in which easy values can be obtained. This table is also another method to find the values of “log” for a given value at a given base. Graph of “log” function touches infinity at zero