Question

Use the laws of logarithms to expand each expression.

43.

(a) \({\log _{10}}\left( {{x^2}{y^3}z} \right)\)

(b) \(\ln \left( {\frac{{{x^4}}}{{\sqrt {{x^2} - 4} }}} \right)\)

Short answer

a) The expression of \({\log _{10}}\left( {{x^2}{y^3}z} \right)\) is \(2{\log _{10}}x + 3{\log _{10}}y + {\log _{10}}z\).

b) The expression of \(\ln \left( {\frac{{{x^4}}}{{\sqrt {{x^2} - 4} }}} \right)\) is \(4\ln x - \,\frac{1}{2}\ln \left( {x + 2} \right) + \frac{1}{2}\ln \left( {x - 2} \right)\).

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:

\(\begin{aligned}{\log _{10}}\left( {{x^2}{y^3}z} \right) &= {\log _{10}}{x^2} + {\log _{10}}{y^3} + {\log _{10}}z\,\,\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,1} \right)\\ &= 2{\log _{10}}x + 3{\log _{10}}y + {\log _{10}}z\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,3} \right)\end{aligned}\)

Thus, the expression of \({\log _{10}}\left( {{x^2}{y^3}z} \right)\) is \(2{\log _{10}}x + 3{\log _{10}}y + {\log _{10}}z\).

b)

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

\(\begin{aligned}\ln \left( {\frac{{{x^4}}}{{\sqrt {{x^2} - 4} }}} \right) &= \ln {x^4} - \ln {\left( {{x^2} - 4} \right)^{\frac{1}{2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,2} \right)\\ &= 4\ln x - \frac{1}{2}\ln \left( {\left( {x + 2} \right)\left( {x - 2} \right)} \right)\,\,\,\,\,\,\,\,\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,3} \right)\\ &= 4\ln x - \,\frac{1}{2}\left( {\ln \left( {x + 2} \right) + \ln \left( {x - 2} \right)} \right)\,\,\,\left( {{\mathop{\rm Law}\nolimits} \,\,1} \right)\\ &= 4\ln x - \,\frac{1}{2}\ln \left( {x + 2} \right) + \frac{1}{2}\ln \left( {x - 2} \right)\end{aligned}\)

Thus, the expression of \(\ln \left( {\frac{{{x^4}}}{{\sqrt {{x^2} - 4} }}} \right)\) is \(4\ln x - \,\frac{1}{2}\ln \left( {x + 2} \right) + \frac{1}{2}\ln \left( {x - 2} \right)\).