Question

Write the following logarithmic function in terms log, 4 (1)/(2) log(2)(x) - 8log(2)(y) - 5l.

Answer 1

The question asks to express a logarithmic function in terms of log base 4. By utilizing the properties of logarithms, such as the product of exponents and the difference of logs, the function can be simplified and the base converted. The resulting expression is log base 4 of
`(x^1/2/y^8)`divided by log base 4 of 2.

Step-by-step explanation:

The question asks to rewrite the logarithmic function in terms of log base 4, which is represented as log₄. Starting with the given function (1)/(2) log(2)(x) - 8log(2)(y), we can apply logarithmic rules to combine the terms. Using the properties of logarithms, specifically that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number
`(log(a^x) = x log(a))`, and that the logarithm of a division is the difference between the logarithms (log(a/b) = log(a) - log(b)), allows for this simplification.

Here's how you convert the given function step by step:

• Multiplying the log by its coefficient: log(2)(x)¹⁻½ - log(2)(y⁸)
• Applying the rule for logarithms of products: log(2)(x¹⁻½/y⁸)

However, we need to change the base from 2 to 4. The change of base formula for logarithms is log_b(a) = log_k(a) / log_k(b), where k is the new base we're working with, in this case, 4. Applying this, we get log₄(x¹⁻½/y⁸) / log₄(2).