Question

Which expression is equivalent to log subscript 12 baseline startfraction x superscript 4 baseline startroot x cubed minus 2 endroot over (x 1) superscript 5 baseline endfraction? 4 log subscript 12 baseline x one-half log subscript 12 baseline (x cubed minus 2) minus 5 log subscript 12 baseline (x times 1) 4 log subscript 12 baseline x one-half log subscript 12 baseline startfraction x cubed over 2 endfraction minus 5 log subscript 12 baseline 1 log subscript 12 baseline 4 x one-half log subscript 12 baseline (x cubed minus 2) minus 5 log subscript 12 baseline (x) 1 4 log subscript 12 baseline x one-half log subscript 12 baseline (x cubed minus 2) minus 5 log subscript 12 baseline (x 1)

Answer 1

There are four basic properties of logarithms. The solution of the logarithmic problem $\rm log_{12}\dfrac{x^4\sqrt{x^3-2}}{(x+1)^5}$ is $\rm 4log_{12}\ x+\dfrac12 log_{12} (x^3-2)-5log_{12}{(x+1)$.

What are the Properties of logarithms?

There are four basic properties of logarithms:

$\rm log_aU+ log_aV = log_a(UV)\\\\\rm log_aU - log_aV = log_a(\dfrac{U}{V})\\\\\rm log_aU^n = n\ log_aU\\\\\rm log_ab = \dfrac{log_xb}{log_xa}$

The given logarithmic problem can be solved in the following manner,

$\rm log_{12}\dfrac{x^4\sqrt{x^3-2}}{(x+1)^5}$

By using the division law of logarithm, $\rm log_a(\dfrac{x}{y}) = log_ax - log_ay$

$\rm log_{12}\dfrac{x^4\sqrt{x^3-2}}{(x+1)^5} = log_{12}\ {x^4\sqrt{x^3-2}}-log_{12}{(x+1)^5}$

By addition law of Logarithm, $\rm log_a(xy)= log_ax+ log_ay$

$\rm log_{12}\ x^4+ log_{12} \sqrt{x^3-2}}-log_{12}{(x+1)^5}$

Using the logarithmic Property, $\rm log\ a^m =mlog\ a$

$=\rm log_{12}\ x^4+ log_{12} \sqrt{x^3-2}}-log_{12}{(x+1)^5}\\\\=log_{12}\ x^4+ log_{12} (x^3-2)^{\frac12}-log_{12}{(x+1)^5$

$=4log_{12}\ x+\dfrac12 log_{12} (x^3-2)-5log_{12}{(x+1)$

Hence, the solution of the logarithmic problem $\rm log_{12}\dfrac{x^4\sqrt{x^3-2}}{(x+1)^5}$ is $\rm 4log_{12}\ x+\dfrac12 log_{12} (x^3-2)-5log_{12}{(x+1)$

Learn more about Logarithms:

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