Question

Solve the equation ${49}^x=7^{x^2-15}$. Check your solution.

Short answer

The solution for the equation is $x=-3,5$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is ${49}^x=7^{x^2-15}$.

Step 2. Simplify the equation.

The given equation ${49}^x=7^{x^2-15}$is solved as:

$\begin{array}{l}{49}^x=7^{x^2-15}\mathrm{....}\left(\text{Given}\right)\\ {\left(7^2\right)}^x={\left(7\right)}^{x^2-15}\\ {\left(7\right)}^{2x}={\left(7\right)}^{x^2-15}\\ 2x=x^2-15\\ x^2-2x-15=0\\ \left(x-5\right)\left(x+3\right)=0\\ x=-3,5\end{array}$

Step 3. Verify the result for obtained value.

Plugging $x=-3,5$in given equation and verify the result.

$\begin{array}{l}{49}^x=7^{x^2-15}\mathrm{....}\left(\text{Given}\right)\\ {49}^{\left(-3\right)}=7^{{\left(-3\right)}^2-15}\mathrm{....}\left(\text{For}x=-3\right)\\ {49}^{\left(-3\right)}=7^{-6}\\ {\left(7^2\right)}^{\left(-3\right)}=7^{-6}\\ 7^{-6}=7^{-6}\mathrm{....}\left(\text{Truefor}x=-3\right)\end{array}$

Similarly, for $x=5$,

$\begin{array}{l}{49}^x=7^{x^2-15}\mathrm{....}\left(\text{Given}\right)\\ {49}^{\left(5\right)}=7^{{\left(5\right)}^2-15}\mathrm{....}\left(\text{For}x=5\right)\\ {49}^{\left(5\right)}=7^{10}\\ {\left(7^2\right)}^5=7^{10}\\ 7^{10}=7^{10}\mathrm{....}\left(\text{Truefor}x=5\right)\end{array}$

Hence the solution is verified for $x=-3,5$.

Step 4. Conclusion.

The solution for the given equation is $x=-3,5$.