Question

The number of solutions of the equation: ${\log }_4\left(x^2-2x\right)={\log }_4(5x-12)$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (C)

$2$

Recall the property that says if ${\log }_{b}x={\log }_{b}y$ then $x=y$

Hence,

$x^2-2x=5x-12$

$\Rightarrow x^2-7x+12=0$

$\Rightarrow (x-3)(x-4)=0$

$\Rightarrow x=3,4$

As the final step we need to take each of the numbers from the above step and plug them into the original equation from the problem statement to make sure we don’t end up taking the logarithm of zero or negative numbers!

Here is the checking work for each of the numbers.

$x=3$;

${\log }_4(3^2-2(3\left)\right)={\log }_4\left(5\right(3)-12)$

${\log }_{4}3={\log }_{4}3$.

So, this solution is correct.

$x=4$;

${\log }_4(4^2-2(4\left)\right)={\log }_4\left(5\right(4)-12)$

${\log }_{4}8={\log }_{4}8$.

So, this solution is also correct.

Hence number of solutions of given equation is $2$.