Expand the given logarithmic expression Assume all variable

Expand the given logarithmic expression. Assume all variable expressions represent positive real numbers. When possible, evaluate logarithmic expressions. Do not use a calculator.

log₂ x/4

answer:____

Write the expression as a single logarithm with a coefficient of 1. Assume all variable expressions represent positive real numbers.

1/2 log₈(x + 6) 5 log₈y

Answer:______

Solution

log 2 ( sqrt x) / sqrt 4

sqrt can be expressed as raised to power 1/2

log 2 (x)^1/2 / 4^1/2

applying log properties

1/2 {log 2 (x) - log 2 (4) }

= 1/2 {log 2x - log 4 }

2) 1/2 log 8 ( x+6) - 5 log 8 y

log 8 (x+6) ^ 1/2 - log 8 y^5

= log 8 ( sqrt (x+6) / y^5 )