Chapter EP, Problem 8.2.6EP

To determine

The solution of the equation.

Explanation of Solution

Given information:

The given equation is $4^{2x+5}<8^{x+1}$

Calculations:

Now to solve the exponential equation we have to first make the base of both sides equal.

Then comparing both sides we solve the equation.

According to the property of equality for exponential functions,

  $\text{If, }b>0\text{ and }b\ne 0,\text{ Then }{b}^x=b^y,\text{ if and only if }x=y.$

Also, according to the property of inequality for exponential functions,

  $\text{If, }b>1,\text{ Then }{b}^x>b^y,\text{ if and only if }x>y.$

Now solving the equation,

  $\begin{array}{l}{4}^{2x+5}<8^{x+1}\\ $ {(2)}^{2\times (2x+5)}<{(2)}^{3\times (x+1)}{\text{ Since, 8=2}}^{\text{3}}\&4{\text{=2}}^{\text{2}}$\text{Now the base of both sides are equal}\text{. Therefore according to the property of equality,}\\ \text{we can write,}\\ 2\times (2x+5)<3\times (x+1)\\ \Rightarrow 4x+10<3x+3\\ \Rightarrow 4x-3x<3-10\\ \Rightarrow x<-7\end{array}$

Therefore the solution of the equation is $x<-7$ .