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} -(120)/(2)=(2x)/(2), \\ -60=x\text{.} \end{gathered}`

Answer: x= -60.