Respuesta :
[tex]2 \times log_{5}( {5x}^{3} ) + ( \frac{1}{3} ) \times log_{5}((x {}^{2}) + 6 ) [/tex]
is equals to
[tex] log_{5}(25 {x}^{6} \times \sqrt[3]{x {}^{2} + 6 } ) [/tex]
Answer: The required expression using single logarithm is [tex]\log_5\{25x^6(x^2+6)^\frac{1}{3}\}.[/tex]
Step-by-step explanation: We are given to write the following logarithmic expression as a single logarithm :
[tex]L=2\log_5(5x^3)+\dfrac{1}{3}\log_5(x^2+6).[/tex]
We will be using the following properties of logarithms :
[tex](i)~a\log b=\log b^a,\\\\(ii)~\log a+\log b=\log (ab).[/tex]
We have
[tex]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}\}.[/tex]
Thus, the required expression using single logarithm is [tex]\log_5\{25x^6(x^2+6)^\frac{1}{3}\}.[/tex]
