Question

Write as the sum and/or difference of logarithms. Express powers as factors \log _4\left(\sqrt{\frac{mn}{19}}\right). g

Answer 1

`f = \log_(4)\left(\sqrt{(m\cdot n)/(19)}\right)` is equivalent to
`f = 0.5\cdot \log_(4) m + 0.5\cdot \log_(4)n -0.5\cdot \log_(4)19`.

Explanation:

Let be
`f = \log_(4)\left(\sqrt{(m\cdot n)/(19)}\right)`, we transform this into an equivalent expression with sums and differences of logarithms by applying logarithm properties:

1)
`\log_(4)\left(\sqrt{(m\cdot n)/(19)}\right)` Given.

2)
`\log_(4)\left[\left((m\cdot n)/(19) \right)^(0.5)\right]` Definition of square root.

3)
`0.5\cdot \log_(4)\left((m\cdot n)/(19) \right)`
`\log_(a) b^(c) = c\cdot \log_(a) b`

4)
`0.5\cdot (\log_(4)m\cdot n -\log_(4) 19)`
`\log_(a) (b)/(c) = \log_(a) b - \log_(a) c`

5)
`0.5\cdot \log_(4) m\cdot n -0.5\cdot \log_(4) 19` Distributive property.

6)
`0.5\cdot (\log_(4)m + \log_(4)n)-0.5\cdot \log_(4)19`
`\log_(a) b\cdot c = \log_(a)b +\log_(a) c`

7)
`0.5\cdot \log_(4) m + 0.5\cdot \log_(4)n -0.5\cdot \log_(4)19` Distributive property/Result.

`f = \log_(4)\left(\sqrt{(m\cdot n)/(19)}\right)` is equivalent to
`f = 0.5\cdot \log_(4) m + 0.5\cdot \log_(4)n -0.5\cdot \log_(4)19`.