What is the exponential form of [tex]$x = \log_4 \frac{y}{5}$[/tex]?

A. [tex]y = 5 \cdot 4^x[/tex]
B. [tex]y = 4 \cdot 5^x[/tex]
C. [tex]y = x^{20}[/tex]
D. [tex]y = 20^x[/tex]



Answer :

To convert the logarithmic equation [tex]\( x = \log_4 \frac{y}{5} \)[/tex] to its exponential form, follow these steps:

1. Start with the given logarithmic equation:
[tex]\[ x = \log_4 \frac{y}{5} \][/tex]

2. Recall the definition of a logarithm. If [tex]\( x = \log_b a \)[/tex], then [tex]\( b^x = a \)[/tex]. Using this definition, we can rewrite the equation in its exponential form. Here, [tex]\( b = 4 \)[/tex] and [tex]\( a = \frac{y}{5} \)[/tex].

3. Therefore, we convert the logarithmic form to exponential form:
[tex]\[ 4^x = \frac{y}{5} \][/tex]

4. To solve for [tex]\( y \)[/tex], multiply both sides of the equation by 5:
[tex]\[ y = 5 \cdot 4^x \][/tex]

So, the exponential form of the given logarithmic equation [tex]\( x = \log_4 \frac{y}{5} \)[/tex] is:
[tex]\[ y = 5 \cdot 4^x \][/tex]

Hence, the correct answer is:
[tex]\[ y = 5 \cdot 4^x \][/tex]