Question

Solve the following logarithms. Write your solution in exact form, simplified if possible.

Answer 1

Recall that:

$\log _bx=a\text{ if and only if }x=b^a.$

Therefore:

$\log _5(5-2x)=3\text{ if and only if }5-2x=5^3.$

Simplifying the above equation we get:

$5-2x=125.$

Adding 2x to the above equation we get:

$\begin{gathered} 5-2x+2x=125+2x, \\ 5=125+2x, \end{gathered}$

Subtracting 125 from the above equation we get:

$\begin{gathered} 5-125=125+2x-125, \\ -120=2x\text{.} \end{gathered}$

Finally, dividing by 2 we get:

$\begin{gathered} -\frac{120}{2}=\frac{2x}{2}, \\ -60=x\text{.} \end{gathered}$

Answer: x= -60.