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cricket20 [7]
3 years ago
14

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

Mathematics
2 answers:
coldgirl [10]3 years ago
8 0

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 }  )
svetlana [45]3 years ago
4 0

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}\}.

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