Question

What is the value of the common logarithm $\log 10000$ ?

Answer 1

Hint: First, we will understand what the logarithmic operator represents in mathematics. A logarithm function or log operator is used when we have to deal with the powers and base of a number, to understand it better which is $\log {x^m} = m\log x$
And also, from the given they ask us to find the common value of the given logarithm function.

Complete step by step answer:
Since in the logarithm $ln$ is $e$ base function and $\log $ is $10$ base function. Here from the given that we have $\log $ so the base is $10$
To solve the given logarithm, we need to know about its property; ${\log _a}a = 1$ and $\log {a^b} = b\log a$
Since $10000$ can be written as in the form of power to $4$ as $10000 = {10^4}$
So, we will convert the given value $10000 = {10^4}$ and substitute the value into the logarithm we get $\log 10000 = \log {10^4}$
Thus, using the logarithm property that $\log {a^b} = b\log a$ and here $a = 10,b = 4$ and hence we get $\log {10^4} = 4\log 10$
Since from the given that we have $\log $ so the base is $10$ and thus we have $4{\log _{10}}10$ and we can now use the property ${\log _a}a = 1$ where $a = 10$ and thus we get $4{\log _{10}}10 = 4 \times 1 \Rightarrow 4$
Hence the value of the common logarithm $\log 10000$ is $4$

Note:
The other more general properties of the logarithm are
 ${\log _{{a^n}}}b = \dfrac{1}{n}{\log _a}b$ where n is any number that we can choose as the base of the log
$\log (\dfrac{a}{b}) = \log a - \log b$
The log rules can be used for the fast exponent calculation using the multiplication operations. The most general base of the log is $10$ and logarithm $ln$ is $e$ base.
As we can see from the basic log formulas then these types of questions are very easy to solve using its property.
The only thing to remember is the difference between them $ln,\log $ because base values change if we apply the wrong base then we get the wrong answer.