Answer: [tex]log_5(25x^6\sqrt[3]{ x^2+6})[/tex]
Step-by-step explanation:
Given the following expression:
[tex]2log_5(5x^3)+\frac{|}{3}log_5(x^2+6)[/tex]
You need to remember the following properties for Logarithms:
[tex]1.\ log(a)+log(b)=log(ab)\\\\2.\ log(a)-log(b)=log(\frac{a}{b})\\\\3.\ log(a)^n=nlog(a)[/tex]
And the following property for Radicals:
[tex]a^{\frac{1}{n}}=\sqrt[n]{a}[/tex]
According to the Power of a power property:
[tex](a^m)^n=a^{mn}[/tex]
Then, you can follow these steps:
Step 1: Apply the third property for logarithms shown above:
[tex]=log_5(5x^3)^2+log_5(x^2+6)^{\frac{1}{3}}[/tex]
Step 2: Apply the Power of a power property:
[tex]=log_5(25x^6)+log_5(x^2+6)^{\frac{1}{3}}[/tex]
Step 3: Using the property for Radicals shown before:
[tex]=log_5(25x^6)+log_5(\sqrt[3]{ x^2+6})[/tex]
Step 4: Now you must apply the first property for logarithms:
[tex]=log_5(25x^6\sqrt[3]{ x^2+6})[/tex]
