Question

What is 2 log Subscript 5 Baseline (5 x cubed) one-third log Subscript 5 Baseline (x squared 6) written as a single logarithm? log Subscript 5 Baseline (25 x Superscript 5 Baseline) RootIndex 3 StartRoot x squared 6 EndRot log Subscript 5 Baseline (25 x Superscript 6 Baseline) RootIndex 3 StartRoot x squared 6 EndRoot log Subscript 5 Baseline 25 x Superscript 5 Baseline log Subscript 5 Baseline 2 log one-third x squared (log Subscript 5 Baseline 25 log Subscript 5 Baseline Superscript 6 Baseline) (log Subscript 5 Baseline RootIndex 3 StartRoot x squared 6 EndRoot).

Answer 1

The solution of the logarithms equation is $\rm log_5(25x^6\sqrt[3]{x^2+6}})$

Given

The following expression:

$\rm 2log_5(5x^3)+\dfrac{1}{3}log_5(x^2+6)$

What properties for Logarithms are used to solve the equation?

The following properties are used in the logarithms equation given below.

$\rm loga+logb=logab\\\\ loga-logb=log(\dfrac{a}{b})\\\\ loga^n=nloga$

According to the Power of a power property:

• Step 1: Apply the third property for logarithms shown above:

                   $\rm 2log_5(5x^3)+\dfrac{1}{3}log_5(x^2+6)\\\\\rm log_5(5x^3)^2+\log_5(x^2+6)^{\frac{1}{3}}$

• Step 2: Apply the Power of a power property:

                     $\rm log_5(5x^3)^2+\log_5(x^2+6)^{\frac{1}{3}}\\\\log_5(25x^6)+log_5(\sqrt[3]{x^2+6}})$

• Step 3: Using the property for Radicals;

                     $\rm log_5(25x^6)+log_5(\sqrt[3]{x^2+6}})\\\\ log_5(25x^6\sqrt[3]{x^2+6}})$

Hence, the solution of the logarithms equation is $\rm log_5(25x^6\sqrt[3]{x^2+6}})$.

To know more about logarithms properties click the link given below.

brainly.com/question/26053315