Final answer:
To expand and simplify logarithms, we use rules such as the logarithm of a product being the sum of logarithms and the logarithm of a number raised to a power being the product of that power and the logarithm.
Step-by-step explanation:
The task at hand is to expand logarithms and rewrite them using their properties. This involves using several rules of logarithms related to products, quotients, and powers.
a) Expand log₃(2x):
To expand log₃(2x), we use the property that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Therefore:
log₃(2x) = log₃(2) + log₃(x)
b) Rewrite ln(x²) as a sum of logarithms:
Using the power rule for logarithms, which states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we can rewrite the expression:
ln(x²) = 2 · ln(x)
c) Expand log₂(4x³):
First, we apply the product rule to separate the constant from the variable:
log₂(4x³) = log₂(4) + log₂(x³)
Then, we use the power rule to bring down the exponent:
log₂(4) + log₂(x³) = log₂(4) + 3 · log₂(x)
d) Simplify log₄(√(2)):
Since logarithms are exponents, we can express the square root as a power of 1/2:
log₄(√(2)) = log₄(2^(1/2))
Applying the power rule:
log₄(2^(1/2)) = (1/2) · log₄(2)