What is expand log_{10}(sqrt(t^9)) ?

The solution to expand log_{10}(sqrt(t^9)) is 9/2 log_{10}(t)

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expand log_{10}(sqrt(t^9))

expand

lo g_{10} (t^{9})

\frac{9}{2} lo g_{10} (t)

Solution steps

lo g_{10} (t^{9})

t^{9} = t^{4} t

= lo g_{10} (t^{4} t)

Apply log rule:

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

lo g_{10} (t^{4} t) = lo g_{10} (t) + lo g_{10} (t^{4})

= lo g_{10} (t) + lo g_{10} (t^{4})

Apply log rule:

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

lo g_{10} (t) + lo g_{10} (t^{4}) = lo g_{10} (t^{4} t)

= lo g_{10} (t t^{4})

Apply log rule:

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

lo g_{10} (t t^{4}) = lo g_{10} (t^{4}) + lo g_{10} (t)

= lo g_{10} (t^{4}) + lo g_{10} (t)

Simplify

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

= lo g_{10} (t^{4}) + \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

= 4 lo g_{10} (t) + \frac{1}{2} lo g_{10} (t)

Factor out common term

lo g_{10} (t)

= lo g_{10} (t) (4 + \frac{1}{2})

4 + \frac{1}{2} = \frac{9}{2}

= \frac{9}{2} lo g_{10} (t)

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What is expand log_{10}(sqrt(t^9)) ?
The solution to expand log_{10}(sqrt(t^9)) is 9/2 log_{10}(t)