Instruction: express each logarith in terms of log₂ M and log₂ N
[tex]1.) \: log2 \: {m}^{6} n[/tex]
[tex]2.) \: log2 \: \sqrt{mn} [/tex]
[tex]3.) \: log2 \frac{ \sqrt{mn} }{ {n}^{3} } [/tex]
[tex]4.) \: log2 ( \frac{m} {n})^{4} [/tex]

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KizunaHighgo KizunaHighgo · Answer 1

1. log₂ m⁶n

If logₙ (xy), then logₙ (xy) = logₙ (x) + logₙ (y)

log₂ (m⁶) + log₂ (n)

If logₙ (xᵃ), then logₙ (xᵃ) = alogₙ (x)

log₂ (m⁶) + log₂ (n) = 6log₂ (m) + log₂ (n)

2. log₂ √mn

If logₙ (xy), then logₙ (xy) = logₙ (x) + logₙ (y)

log₂ √m + log₂ √n

Any value in a radical symbol assuming that the root is 2 have always the exponent of ½. So √x = x¹/²

log₂ (m¹/²) + log₂ (n¹/²)

If logₙ (xᵃ), then logₙ (xᵃ) = alogₙ (x)

log₂ m¹/² + log₂ n¹/² = ½log₂ (m) + ½log₂ (n)

3. log₂ √mn / n³

If logₙ (x/y), then logₙ (x/y) = logₙ (x) - logₙ (y)

log₂ (√mn) - log₂ (n³)

Any value in a radical symbol assuming that the root is 2 have always the exponent of ½. So √x = x¹/²

log₂ (m¹/²) + log₂ (n¹/²) - log₂ (n³)

If logₙ (xᵃ), then logₙ (xᵃ) = alogₙ (x)

log₂ (m¹/²) + log₂ (n¹/²) - log₂ (n³) = ⅓log₂ (m) + ½log₂ (n) - 3log₂ (n)

4. log₂ (m/n)⁴

If log (x/y)ᵃ then log (x/y)ᵃ = log (xᵃ/yᵃ)

log₂ (m⁴/n⁴)

If logₙ (x/y), then logₙ (x/y) = logₙ (x) - logₙ (y)

log₂ (m⁴) - log₂ (n⁴)

If logₙ (xᵃ), then logₙ (xᵃ) = alogₙ (x)

log₂ (m⁴) - log₂ (n⁴) = 4log₂ (m) - 4log₂ (n)

[tex] \: [/tex]

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