Alright, let's solve the equation step-by-step.
We start with the given equation:
[tex]\[ \log_5(y) - 4 \log_5(2) = 2 \][/tex]
First, we recognize that we can use logarithmic properties to simplify the equation. One useful property is that [tex]\( n \log_b(x) = \log_b(x^n) \)[/tex]. Let's apply this property to the term [tex]\( 4 \log_5(2) \)[/tex]:
[tex]\[ \log_5(y) - \log_5(2^4) = 2 \][/tex]
Since [tex]\( 2^4 = 16 \)[/tex], the equation then becomes:
[tex]\[ \log_5(y) - \log_5(16) = 2 \][/tex]
Next, we apply another logarithmic property: [tex]\( \log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right) \)[/tex]. So we have:
[tex]\[ \log_5\left(\frac{y}{16}\right) = 2 \][/tex]
To solve for [tex]\( y \)[/tex], we convert the logarithmic equation into its exponential form. Recall that [tex]\( \log_b(a) = c \)[/tex] is equivalent to [tex]\( b^c = a \)[/tex]. Therefore, we have:
[tex]\[ 5^2 = \frac{y}{16} \][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[ 25 = \frac{y}{16} \][/tex]
Multiply both sides of the equation by 16:
[tex]\[ y = 25 \cdot 16 \][/tex]
Simplify the right-hand side:
[tex]\[ y = 400 \][/tex]
Thus, the solution to the equation [tex]\( \log_5(y) - 4 \log_5(2) = 2 \)[/tex] is:
[tex]\[ y = 400 \][/tex]
