What is expand log_{10}(sqrt(t^5)) ?

The solution to expand log_{10}(sqrt(t^5)) is 5/2 log_{10}(t)

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expand log_{10}(sqrt(t^5))

expand

lo g_{10} (t^{5})

\frac{5}{2} lo g_{10} (t)

Solution steps

lo g_{10} (t^{5})

t^{5} = t^{2} t

= lo g_{10} (t^{2} t)

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

lo g_{10} (t^{2} t) = lo g_{10} (t) + lo g_{10} (t^{2})

= lo g_{10} (t) + lo g_{10} (t^{2})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

lo g_{10} (t) + lo g_{10} (t^{2}) = lo g_{10} (t^{2} t)

= lo g_{10} (t t^{2})

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

lo g_{10} (t t^{2}) = lo g_{10} (t^{2}) + lo g_{10} (t)

= lo g_{10} (t^{2}) + lo g_{10} (t)

Simplify

lo g_{10} (t) : \frac{1}{2} lo g_{10} (t)

= lo g_{10} (t^{2}) + \frac{1}{2} lo g_{10} (t)

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= 2 lo g_{10} (t) + \frac{1}{2} lo g_{10} (t)

Factor out common term

lo g_{10} (t)

= lo g_{10} (t) (2 + \frac{1}{2})

2 + \frac{1}{2} = \frac{5}{2}

= \frac{5}{2} lo g_{10} (t)

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What is expand log_{10}(sqrt(t^5)) ?
The solution to expand log_{10}(sqrt(t^5)) is 5/2 log_{10}(t)