Answer :
To rewrite the expression [tex]\(\log(20) - \log(5)\)[/tex] in the form [tex]\(\log(c)\)[/tex], we can use one of the properties of logarithms. The specific property we will use is:
[tex]\[ \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \][/tex]
Using this property, we can transform the given expression. Here are the steps:
1. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the expression [tex]\(\log(20) - \log(5)\)[/tex]. In this case, [tex]\(a = 20\)[/tex] and [tex]\(b = 5\)[/tex].
2. Apply the logarithm property to combine the two logarithms into one:
[tex]\[ \log(20) - \log(5) = \log\left(\frac{20}{5}\right) \][/tex]
3. Simplify the fraction inside the logarithm:
[tex]\[ \frac{20}{5} = 4 \][/tex]
4. Substitute the simplified fraction back into the logarithm:
[tex]\[ \log\left(\frac{20}{5}\right) = \log(4) \][/tex]
So, the expression [tex]\(\log(20) - \log(5)\)[/tex] can be rewritten in the form [tex]\(\log(c)\)[/tex] as:
[tex]\[ \log(20) - \log(5) = \log(4) \][/tex]
Thus, [tex]\(c = 4\)[/tex].
[tex]\[ \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \][/tex]
Using this property, we can transform the given expression. Here are the steps:
1. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the expression [tex]\(\log(20) - \log(5)\)[/tex]. In this case, [tex]\(a = 20\)[/tex] and [tex]\(b = 5\)[/tex].
2. Apply the logarithm property to combine the two logarithms into one:
[tex]\[ \log(20) - \log(5) = \log\left(\frac{20}{5}\right) \][/tex]
3. Simplify the fraction inside the logarithm:
[tex]\[ \frac{20}{5} = 4 \][/tex]
4. Substitute the simplified fraction back into the logarithm:
[tex]\[ \log\left(\frac{20}{5}\right) = \log(4) \][/tex]
So, the expression [tex]\(\log(20) - \log(5)\)[/tex] can be rewritten in the form [tex]\(\log(c)\)[/tex] as:
[tex]\[ \log(20) - \log(5) = \log(4) \][/tex]
Thus, [tex]\(c = 4\)[/tex].
