Question

What is written as a 2log5(5x^3)+(1/3)log5((x^2)+6) single logarithm?

Answer 1

$2 \times log_{5}( {5x}^{3} ) + ( \frac{1}{3} ) \times log_{5}((x {}^{2}) + 6 )$

is equals to
$log_{5}(25 {x}^{6} \times \sqrt[3]{x {}^{2} + 6 } )$

Answer 2

Answer:  The required expression using single logarithm is $\log_5\{25x^6(x^2+6)^\frac{1}{3}\}.$

Step-by-step explanation:  We are given to write the following logarithmic expression as a single logarithm :

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

We will be using the following properties of logarithms :

$(i)~a\log b=\log b^a,\\\\(ii)~\log a+\log b=\log (ab).$

We have

$L\\\\=2\log_5(5x^3)+\dfrac{1}{3}\log_5(x^2+6)\\\\=\log_5(5x^3)^2+\log_5(x^2+6)^\frac{1}{3}\\\\=\log_5\{(5x^3)^2\times (x^2+6)^\frac{1}{3}\}\\\\=\log_5\{25x^6(x^2+6)^\frac{1}{3}\}.$

Thus, the required expression using single logarithm is $\log_5\{25x^6(x^2+6)^\frac{1}{3}\}.$