Answer: =[tex]log_5\ (25\ x^6\ \sqrt[3]{x^2+6})[/tex]
Step-by-step explanation:
Given log expression [tex]2log_5(5x^3)+\frac{1}{3} log_5((x^2)+6)[/tex].
First we would apply log rule of exponents : [tex]nlog_b(a) = log_b(a)^n[/tex]
[tex]2log_5(5x^3)+\frac{1}{3} log_5((x^2)+6) = log_5(5x^3)^2+ log_5((x^2)+6)^{\frac{1}{3}}[/tex].
Converting rational exponent into radical form, we get
[tex]log_5(5x^3)^2+ log_5\sqrt[3]{((x^2)+6)}[/tex].
Simplifying [tex](5x^3)^2 = 5^2x^{3\times 2} = 25x^6[/tex]
= [tex]log_5(25x^6)+ log_5\sqrt[3]{((x^2)+6)}[/tex].
Applying sum of logs rule [tex]log_b(m)+log_b(n) = log_b(m\times n)[/tex].
=[tex]log_5\ (25\ x^6\ \sqrt[3]{x^2+6})[/tex]
