Chapter 6.5, Problem 44E

To determine

To solve the equation:

  $3^{4x+3}={81}^x$

Answer to Problem 44E

No solution.

Explanation of Solution

Given:

  $3^{4x+3}={81}^x$

Concept Used:

• An equation that is true for all the values of variable has infinitely many solution. And an equation that is not true for any value of the variable has no solution.
• Two powers with the same positive base(not equals to 1), are equal if and only if their exponents are equal, that is:

  $b^x=b^y\iff x=y;\text{ where }b>0,\text{ and }b\ne 1$

Calculation:

To solve the equation:

  $3^{4x+3}={81}^x$

First, simplify the equation so the base of the equation is same, as

  $\begin{array}{c}{3}^{4x+3}={81}^x\\ 3^{4x+3}={\left(3\times 3\times 3\times 3\right)}^x\\ \Rightarrow 3^{4x+3}={\left(3^4\right)}^x\\ \Rightarrow 3^{4x+3}=3^{4x}\end{array}$

Now, since base of above powers are same; thus

  $\begin{array}{c}{3}^{4x+3}=3^{4x}\\ \Rightarrow 4x+3=4x\\ \Rightarrow 4x-4x=-3\\ \Rightarrow 0=-3\end{array}$

But, the statement $0=-3$ is never true. Thus the equation is never true for any value of x .

Thus, the given equation has no solution.