Solve for x log base 5 of x^10- log base 5 of x^6=21

Use the quotient property of logarithms, ${\displaystyle {\log }_b\left(x\right)-{\log }_b\left(y\right)={\log }_b\left(\frac{x}{y}\right)}$.

Cancel the common factor of ${\displaystyle x^{10}}$ and ${\displaystyle x^6}$.

For logarithmic equations, ${\displaystyle {\log }_b\left(x\right)=y}$ is equivalent to ${\displaystyle b^y=x}$ such that ${\displaystyle x>0}$, ${\displaystyle b>0}$, and ${\displaystyle b\ne 1}$. In this case, ${\displaystyle b=5}$, ${\displaystyle x=x^4}$, and ${\displaystyle y=21}$.

Substitute the values of ${\displaystyle b}$, ${\displaystyle x}$, and ${\displaystyle y}$ into the equation ${\displaystyle b^y=x}$.

Rewrite the equation as ${\displaystyle x^4=5^{21}}$.

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

Simplify ${\displaystyle \pm \sqrt[4]{5^{21}}}$.

The complete solution is the result of both the positive and negative portions of the solution.