Answer :
To simplify the expression [tex]\log_5(20) + \log_5\left(\frac{125}{25}\right) - \log_5\left(\frac{1}{25}\right)[/tex], we can use the properties of logarithms.
Use the Quotient Rule:
- The quotient rule states that [tex]\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)[/tex]. This allows us to combine the logarithms in cases where there is a subtraction.
- First, simplify the expression inside the second logarithm: [tex]\frac{125}{25} = 5[/tex].
- Thus, the expression becomes [tex]\log_5(20) + \log_5(5) - \log_5\left(\frac{1}{25}\right)[/tex].
Use the Quotient Rule Again:
- For [tex]-\log_5\left(\frac{1}{25}\right)[/tex], apply the rule to rewrite it as [tex]\log_5(25)[/tex] due to the negative sign.
Apply the Product Rule:
- The product rule states that [tex]\log_b(x) + \log_b(y) = \log_b(xy)[/tex]. Use this to combine the positive terms:
[tex]\log_5(20 \times 5) = \log_5(100).[/tex]
- The product rule states that [tex]\log_b(x) + \log_b(y) = \log_b(xy)[/tex]. Use this to combine the positive terms:
Simplify Further:
- The expression now becomes [tex]\log_5(100) - \log_5(25)[/tex].
- Use the quotient rule again: [tex]\log_5\left(\frac{100}{25}\right)[/tex].
Simplify the Fraction:
- Simplify [tex]\frac{100}{25} = 4[/tex]. Thus, the expression is now [tex]\log_5(4)[/tex].
Therefore, the simplified form of the expression is [tex]\log_5(4)[/tex].