Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • What is the value of x satisfying the inequality <img alt="\\\mathrm{\log_{\;5}(x+5) > \log_{\;7}(x+5)}" src="https://learn.careers360.com/latex-image/?%5C%5C%5Cmathrm%7B%5Clog_%7B%5C%3B5%7

What is the value of x satisfying the inequality \\\mathrm{\log_{\;5}(x+5) > \log_{\;7}(x+5)}?

Option: 1

x> -4


Option: 2

x< -4


Option: 3

x\in(-4,4)


Option: 4

x\leq -4


Answers (1)

best_answer

As we have learnt in 

 

 

Logarithmic Inequalities -

Logarithmic inequalities:

\mathrm{\log_ax>\log_ay=\left\{\begin{matrix} \mathrm{x>y}, &\;\;\mathrm{if\;a>1} \\ \mathrm{x<y}, &\;\;\mathrm{if\;0<a<1} \end{matrix}\right.}

\mathrm{\log_ax>y=\left\{\begin{matrix} \mathrm{x>a^y}, &\;\;\mathrm{if\;a>1} \\ \mathrm{x<a^y}, &\;\;\mathrm{if\;0<a<1} \end{matrix}\right.}

\mathrm{\log_ax<y=\left\{\begin{matrix} \mathrm{x<a^y}, &\;\;\mathrm{if\;a>1} \\ \mathrm{x>a^y}, &\;\;\mathrm{if\;0<a<1} \end{matrix}\right.}

Generally, from these inequalities, we can conclude that logarithmic functions are monotonically increasing for a >1 and decreasing for 0 < a < 1. Basic property which we must remember about log is that argument ( means x or y) must be positive.

-

first, we do the change of base to make the base of both side same

{\color{Blue} \log_ b x = \frac{\log_a x}{\log_a b}}

\\\mathrm{\frac{\log(x+5)}{log\; 5} > \frac{\log(x+5)}{\log7}}

Now for this equation to be true, we observe that 

\\\mathrm{\frac{\log(x+5)}{log\; 5} > \frac{\log(x+5)}{\log7}}\\\frac{\log \left(x+5\right)}{\log \left(5\right)}-\frac{\log {}\left(x+7\right)}{\log7}>0\\\log(x+5)>0\\x+5>10^0\\x+5>1\\x>-4

Posted by

Divya Prakash Singh

View full answer

Similar Questions