Question

If $7^n\times 7^3=7^{12}$, what is the value of $n$?

• A. $2$
• B. $4$
• C. $9$
• D. $15$
• E. $36$

Answer 1

Answer: (C)

$9$

Given, $7^n$ $\times$ $7^3$ $=$ $7^{12}$

Rearranging terms, we get

$7^n$ $=$ $\frac{7^{12}}{7^3}$

We know that

$\frac{x^a}{x^b}$ $=$ $x^{a-b}$

Hence, $7^n$ $=$ $7^{12-3}$

$\therefore 7^n$ $=$ $7^9$

Again rearranging terms, we get

$n$ $=$ ${\log }_7{7}^9$

We know that

${\log }_x{x}^a$ $=$ $a$${\log }_{x}x$ and ${\log }_{x}x$ $=$ $1$

Hence, $n$ $=$ $9$${\log }_{7}7$

$\therefore n$ $=$ $9$ $\times$ $1$

$\therefore n$ $=$ $9$

Therefore, the value of ${}^{\prime }{n}^{\prime }$ is ${}^{\prime }{9}^{\prime }$.