Answer :
To solve the equation [tex]\log_5 p - \log_5 4 = 2[/tex], we begin by using a property of logarithms known as the quotient rule. The quotient rule states that [tex]\log_b (a) - \log_b (c) = \log_b \left( \frac{a}{c} \right)[/tex].
Applying this rule to the given equation, we have:
[tex]\log_5 \left( \frac{p}{4} \right) = 2[/tex]
The next step is to convert the logarithmic equation to an exponential form. Recall that if [tex]\log_b (a) = c[/tex], then [tex]a = b^c[/tex]. Applying this to our equation:
[tex]\frac{p}{4} = 5^2[/tex]
Calculate [tex]5^2[/tex]:
[tex]5^2 = 25[/tex]
So the equation becomes:
[tex]\frac{p}{4} = 25[/tex]
To solve for [tex]p[/tex], multiply both sides by 4:
[tex]p = 25 \times 4[/tex]
Calculate the right side:
[tex]p = 100[/tex]
Therefore, the value of [tex]p[/tex] is [tex]100[/tex]. This process uses logarithmic properties and algebraic manipulation to find the solution.