To understand the relationship implied by the logarithmic equation [tex]\( y = \log_z(x) \)[/tex], let's explore the definition of logarithms.
The equation [tex]\( y = \log_z(x) \)[/tex] represents the logarithm of [tex]\( x \)[/tex] with base [tex]\( z \)[/tex]. This means that [tex]\( y \)[/tex] is the exponent to which the base [tex]\( z \)[/tex] must be raised to produce [tex]\( x \)[/tex].
Let's express this relationship mathematically:
1. Considering the definition of logarithms, we can write:
[tex]\[
y = \log_z(x)
\][/tex]
2. By definition, this implies:
[tex]\[
z^y = x
\][/tex]
Hence,
[tex]\[ z^y = x \][/tex]
Comparing this result with the given options:
(a) [tex]\( x^y = z \)[/tex] — This statement is incorrect based on our derivation.
(b) [tex]\( z^y = x \)[/tex] — This statement is correct and matches our derivation.
(c) [tex]\( x^2 = y \)[/tex] — This statement is incorrect as it does not relate to the logarithmic definition.
(d) [tex]\( y^2 = x \)[/tex] — This statement is also incorrect as it does not relate to the logarithmic definition.
Thus, the correct implication of [tex]\( y = \log_z(x) \)[/tex] is:
(b) [tex]\( z^y = x \)[/tex]
