How to solve log_(y)(5y) = 2

Question

How to solve
`log_(y)(5y) = 2`

Answer 1

y = 5

Explanation:

Expand the logarithm:

`\log_y{(5)}+\log_y{(y)}=2\\\\\frac{\log{(5)}}{\log{(y)}}+1=2 \quad\text{change of base formula}\\\\\frac{\log{(5)}}{\log{(y)}}=1 \quad\text{subtract 1}\\\\\log{(5)}=\log{(y)} \quad\text{multiply by log(y)}\\\\5=y \quad\text{take the anti-log}`

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You can also take the antilog first:

5y = y²

y(y -5) = 0 . . . . . subtract 5y, factor

y = 0 or 5 . . . . . y=0 is not a viable solution, so y=5.