High School

What is the exponential form of the equation [tex]z = \log_{54} y[/tex]?

A. [tex]94 = z^r[/tex]
B. [tex]94^z = y[/tex]
C. [tex]94^y = z[/tex]
D. [tex]94 = y^2[/tex]

Answer :

To convert the logarithmic equation [tex]\( z = \log_{54} y \)[/tex] into its exponential form, let's follow these steps:

1. Understanding the Logarithmic Equation: The equation [tex]\( z = \log_{54} y \)[/tex] means that [tex]\( z \)[/tex] is the power to which the base 54 must be raised to get [tex]\( y \)[/tex].

2. Converting to Exponential Form: A logarithmic equation of the form [tex]\( z = \log_b a \)[/tex] can be rewritten in exponential form as [tex]\( a = b^z \)[/tex].

3. Apply to Your Equation: In the given equation, [tex]\( z = \log_{54} y \)[/tex], we have:
- The base [tex]\( b = 54 \)[/tex],
- The logarithm result [tex]\( z \)[/tex] is the exponent,
- The number [tex]\( y \)[/tex] is what you get when you exponentiate the base by this logarithm result.

So, this becomes [tex]\( y = 54^z \)[/tex].

4. Match with Given Options: We need to find which option corresponds to [tex]\( y = 54^z \)[/tex]:
- (A) [tex]\( 94 = z^r \)[/tex]
- (B) [tex]\( 94^z = y \)[/tex]
- (C) [tex]\( 94^y = z \)[/tex]
- (D) [tex]\( 94 = y^2 \)[/tex]

None of these options match the given [tex]\( y = 54^z \)[/tex] directly. However, given the modification in problem statement, it seems compatible with the understanding that option (B) [tex]\( 94^z = y \)[/tex] is suggested as a correct shorthand outcome.

It's important to rely on precise details when dealing with math conversions. For problems similar to log, always ensure the base, exponent, and expression are accurately interpreted.