Question

Use the laws of logarithms to expand each expression.

44.

(a) \(\ln \sqrt {\frac{{3x}}{{x - 3}}} \)

(b) \({\log _2}\left[ {\left( {{x^3} + 1} \right)\sqrt[3]{{{{\left( {x - 3} \right)}^2}}}} \right]\)

Short answer

a) The expression of \(\ln \sqrt {\frac{{3x}}{{x - 3}}} \) is \(\frac{1}{2}\ln 3 + \frac{1}{2}\ln x - \frac{1}{2}\ln \left( {x - 3} \right)\).

b) The expression of \({\log _2}\left( {\left[ {{x^3} + 1} \right)\sqrt[3]{{{{\left( {x - 3} \right)}^2}}}} \right]\) is \({\log _2}\left( {{x^3} + 1} \right) + \frac{2}{3}{\log _2}\left( {x - 3} \right)\).

Step-by-step solution

Law of Logarithm

When \(x\) and \(y\) are positive numbers, then;

1. \({\log _b}\left( {xy} \right) = {\log _b}x + {\log _b}y\)
2. \({\log _b}\left( {\frac{x}{y}} \right) = {\log _b}x - {\log _b}y\)
3. \({\log _b}\left( {{x^r}} \right) = r{\log _b}x\) (Where \(r\) is any real number)

Use the laws of logarithms to expand the expression

a)

Use the laws of logarithms to expand the expression as shown below:

\(\begin{aligned}\ln \sqrt {\frac{{3x}}{{x - 3}}} &= \ln {\left( {\frac{{3x}}{{x - 3}}} \right)^{\frac{1}{2}}}\\ &= \frac{1}{2}\ln \left( {\frac{{3x}}{{x - 3}}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,3} \right)\\ &= \frac{1}{2}\left( {\ln 3 + \ln x - \ln \left( {x - 3} \right)} \right)\,\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,1\,\,\,{\mathop{\rm and}\nolimits} \,\,2} \right)\\ &= \frac{1}{2}\ln 3 + \frac{1}{2}\ln x - \frac{1}{2}\ln \left( {x - 3} \right)\end{aligned}\)

Thus, the expression of \(\ln \sqrt {\frac{{3x}}{{x - 3}}} \) is \(\frac{1}{2}\ln 3 + \frac{1}{2}\ln x - \frac{1}{2}\ln \left( {x - 3} \right)\).

b)

Use the laws of logarithms to expand the expression as shown below:

\(\begin{aligned}{\log _2}\left[ {\left( {{x^3} + 1} \right)\sqrt[3]{{{{\left( {x - 3} \right)}^2}}}} \right] &= {\log _2}\left( {{x^3} + 1} \right) + {\log _2}\sqrt[3]{{{{\left( {x - 3} \right)}^2}}}\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,1} \right)\\ &= {\log _2}\left( {{x^3} + 1} \right) + {\log _2}{\left( {x - 3} \right)^{\frac{2}{3}}}\\ &= {\log _2}\left( {{x^3} + 1} \right) + \frac{2}{3}{\log _2}\left( {x - 3} \right)\,\,\,\,\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,3} \right)\end{aligned}\)

Thus, the expression of \({\log _2}\left[ {\left( {{x^3} + 1} \right)\sqrt[3]{{{{\left( {x - 3} \right)}^2}}}} \right]\) is \({\log _2}\left( {{x^3} + 1} \right) + \frac{2}{3}{\log _2}\left( {x - 3} \right)\).