Final answer:
The base expression for log₄ 75 cannot be written as a simple fraction because 75 is not a power of 4. Instead, using properties of logarithms, it can be written as a sum of simpler logarithmic terms: log₄ 3 + 2 × log₄ 5.
Step-by-step explanation:
The question asks to express the logarithm log₄ 75 as a fraction based on the properties of logarithms. Recalling the property of logarithms that logam - logan corresponds to loga(m/n), we use this to rewrite the logarithm in question.
Since 75 is not a power of 4, we cannot express log₄ 75 as a simple fraction. However, we know that logarithms can be written in terms of other bases. In this case, we can write it as:
log₄ 75 = log₄ (3× 25) = log₄ (3× 52) = log₄ 3 + 2 × log₄ 5
While this is not strictly a fraction, it does express log₄ 75 as a sum of fractions where each part is a logarithm of a simpler number relative to the base 4.