Condense the following log into a single log: log_(3) x+(1)/(3)log_(3) y-5log_(3) z

Question

Condense the following log into a single log:


`log_(3) x+(1)/(3)log_(3) y-5log_(3) z`

Answer 1

`log_(3)(x . y^(1)/(3) / z^5 )`

Explanation:

Given in the question an expression

`log_(3) x+(1)/(3)log_(3) y-5log_(3) z`

To Condense this log into a single log we will use logarithm rules

1)Power rule

logb(x^y) = y ∙ logb(x)

`\frac{1}3}log_(3) y = log_(3)y^(1)/(3)`

`5log_(3) z = log_(3) z^5`

2)Product rule

logb(x ∙ y) = logb(x) + logb(y)

`log_(3) x + log_(3)y^(1)/(3) = log_(3)(x . y^(1)/(3) )`

3)qoutient rule

`logb(x / y) = logb(x) - logb(y)\\log_(3)(x . y^(1)/(3) )- log_(3) z^5`

=
`log_(3)(x . y^(1)/(3) / z^5 )`