Question

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

Answer 1

`2 * log_(5)( {5x}^(3) ) + ( (1)/(3) ) * log_(5)((x {}^(2)) + 6 )`

is equals to

`log_(5)(25 {x}^(6) * \sqrt[3]{x {}^(2) + 6 } )`

Answer 2

Answer: The required expression using single logarithm is
`\log_5\{25x^6(x^2+6)^(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)+(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)+(1)/(3)\log_5(x^2+6)\\\\=\log_5(5x^3)^2+\log_5(x^2+6)^(1)/(3)\\\\=\log_5\{(5x^3)^2* (x^2+6)^(1)/(3)\}\\\\=\log_5\{25x^6(x^2+6)^(1)/(3)\}.`

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