Question

Solve the equation ${10}^{x-1}={100}^{2x-3}$. Check your solution.

Short answer

The solution for the equation is $x=\frac{5}{3}$.

Step-by-step solution

Step 1. Write down the given equation.

The given equation is ${10}^{x-1}={100}^{2x-3}$.

Step 2. Simplify the equation.

The given equation ${10}^{x-1}={100}^{2x-3}$is solved as:

$\begin{array}{l}{10}^{x-1}={100}^{2x-3}\mathrm{....}\left(\text{Given}\right)\\ {\left(10\right)}^{x-1}={\left({10}^2\right)}^{2x-3}\\ {\left(10\right)}^{x-1}={\left(10\right)}^{4x-6}\\ x-1=4x-6\\ 3x=5\\ x=\frac{5}{3}\end{array}$

Step 3. Verify the result for obtained value.

Plugging $x=\frac{5}{3}$in given equation and verify the result.

role="math" localid="1648199179427" $\begin{array}{l}{10}^{x-1}={100}^{2x-3}\mathrm{....}\left(\text{Given}\right)\\ {10}^{\left(\frac{5}{3}\right)-1}={100}^{2\left(\frac{5}{3}\right)-3}\\ {10}^{\frac{2}{3}}={100}^{\frac{10}{3}-3}\\ {10}^{\frac{2}{3}}={100}^{\frac{1}{3}}\\ {10}^{\frac{2}{3}}={\left({10}^2\right)}^{\frac{1}{3}}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $x=\frac{5}{3}$.

Step 4. Conclusion.

The solution for the given equation is $x=\frac{5}{3}$.