Question

Find the number of positive integers which have the characteristic 3, when the base of the log is 7.
(a). 2058
(b). 1029
(c). 1030
(d). 2060

Answer 1

Hint: We have been given that the characteristic 3 and the base of the log is 7, hence the range of logx base 7 will be $\left[ 3,4 \right)$ , from that we have to find the range of x and from that we will the number of integers that lie between that range.

Complete step-by-step answer:

Let’s start solving this question.
The range of ${{\log }_{7}}x$ is $\left[ 3,4 \right)$ which is given in the question.
Now we will find the value of x from this given value of range.
We will be using the formula ${{\log }_{b}}a=x\Rightarrow a={{b}^{x}}$.
Hence, we get
$\begin{aligned} & 3\le {{\log }_{7}}x<4 \\\ & {{7}^{3}}\le x<{{7}^{4}} \\\ & 343\le x<2401 \\\ \end{aligned}$
Hence, we have found the range of value of x between the two integers.
Now we will subtract the two integers to find the number of integers that lie between them.
Hence, we get,
$2401-343=2058$
Hence, the number of positive integers which have the characteristic 3, when the base of the log is 7 is 2058.
Hence, option (a) is correct.

Note: One can also find the value of range of x by using antilog in both the sides of the equation $3\le {{\log }_{7}}x<4$, and after that we will get the same range of x as we have as we have got above.
This formula ${{\log }_{b}}a=x\Rightarrow a={{b}^{x}}$ must be kept in mind.