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You see a 4 and a 8 in the bases. The idea here is to convert them into the same base of 2 first, since both are powers of 2 (4 = 2² and 8 = 2³)

Here is one way you can do this :

log₄y = p

log₂ y / log₂ 4 = p (Change of base)

log₂ y / log₂ 2² = p

log₂ y / 2log₂ 2 = p (bring down the power)

log₂ y / 2 = p

log₂ y = 2p ①


Next,

log₈2y = q

log₈ 2 + log₈ y = q (Addition of logs)

log₈ 8¹/³ + log₂ y / log₂ 8 = q (Change of base)

⅓ log₈ 8 + log₂ y / log₂ 2³ = q

⅓ + log₂ y / 3log₂ 2 = q (bring down the power)

⅓ + log₂ y / 3 = q

1 + log₂ y = 3q ②

Sub ① into ②,

1 + 2p = 3q

2p = 3q - 1

The other way : change the log form back into index form first.



log₄y = p

y = 4ᵖ

y = (2²)ᵖ

y = 2²ᵖ ①


Next,


log₈ 2y = q

2y = 8ᑫ

2y = (2³)ᑫ

2y = 2³ᑫ

y = 2³ᑫ / 2

y = 2³ᑫ⁻¹ ②


① = ②

2²ᵖ = 2³ᑫ⁻¹

Since they are both single terms and they have identical bases, their powers are equal.

2p = 3q - 1

Thank you so much!

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