The logarithm function, often denoted as [tex]\( \log_b(a) \)[/tex], asks for the exponent [tex]\( x \)[/tex] to which the base [tex]\( b \)[/tex] must be raised to obtain the number [tex]\( a \)[/tex].
Here, the question is about the logarithm of unity (which means 1) to any base. Therefore, we need to find:
[tex]\[ \log_b(1) \][/tex]
Step-by-step reasoning:
1. Definition of Logarithm: The general definition of a logarithm states that [tex]\( \log_b(a) = x \)[/tex] means [tex]\( b^x = a \)[/tex].
2. Apply to the Given Problem: We need to determine [tex]\( \log_b(1) \)[/tex]. Set [tex]\( \log_b(1) = x \)[/tex].
3. Translate to Exponential Form: Based on the definition, this translates to the equation [tex]\( b^x = 1 \)[/tex].
4. Analyze the Exponential Equation: For any base [tex]\( b \)[/tex] (assuming [tex]\( b \neq 0 \)[/tex]), the equation [tex]\( b^x = 1 \)[/tex] holds true only when [tex]\( x = 0 \)[/tex]. This is because any non-zero number raised to the power of 0 is 1.
Therefore, the logarithm of 1 to any base is indeed:
[tex]\[ \log_b(1) = 0 \][/tex]
So, the correct answer is (d) 0.
