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?

Answer 1

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

Step-by-step explanation:

Given the following expression:

$2log_5(5x^3)+\frac{|}{3}log_5(x^2+6)$

You need to remember the following properties for Logarithms:

$1.\ log(a)+log(b)=log(ab)\\\\2.\ log(a)-log(b)=log(\frac{a}{b})\\\\3.\ log(a)^n=nlog(a)$

And the following property for Radicals:

$a^{\frac{1}{n}}=\sqrt[n]{a}$

According to the Power of a power property:

$(a^m)^n=a^{mn}$

Then, you can follow these steps:

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

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

Step 2: Apply the Power of a power property:

$=log_5(25x^6)+log_5(x^2+6)^{\frac{1}{3}}$

 Step 3: Using the property for Radicals shown before:

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

Step 4: Now you must apply the first property for logarithms:

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

Answer 2

B

Step-by-step explanation: