What is log_{4}((x^3e^2)/((4x+1)^{1/2)}) ?

The solution to log_{4}((x^3e^2)/((4x+1)^{1/2)}) is 3log_{4}(x)+2log_{4}(e)-1/2 log_{4}(4x+1)

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simplify log_{4}((x^3e^2)/((4x+1)^{1/2)})

simplify

lo g_{4} (\frac{x ^{3} e ^{2}}{( 4 x + 1 ) ^{\frac{1}{2}}})

3 lo g_{4} (x) + 2 lo g_{4} (e) - \frac{1}{2} lo g_{4} (4 x + 1)

Solution steps

lo g_{4} (\frac{x ^{3} e ^{2}}{( 4 x + 1 ) ^{\frac{1}{2}}})

Apply log rule:

lo g_{c} (\frac{a}{b}) = lo g_{c} (a) - lo g_{c} (b)

lo g_{4} (\frac{x ^{3} e ^{2}}{( 4 x + 1 ) ^{\frac{1}{2}}}) = lo g_{4} (x^{3} e^{2}) - lo g_{4} ((4 x + 1)^{\frac{1}{2}})

= lo g_{4} (e^{2} x^{3}) - lo g_{4} ((4 x + 1)^{\frac{1}{2}})

Apply log rule:

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

lo g_{4} ((4 x + 1)^{\frac{1}{2}}) = \frac{1}{2} lo g_{4} (4 x + 1)

= lo g_{4} (x^{3} e^{2}) - \frac{1}{2} lo g_{4} (4 x + 1)

Apply log rule:

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

lo g_{4} (x^{3} e^{2}) = lo g_{4} (x^{3}) + lo g_{4} (e^{2})

= lo g_{4} (x^{3}) + lo g_{4} (e^{2}) - \frac{1}{2} lo g_{4} (4 x + 1)

Apply log rule:

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

lo g_{4} (x^{3}) = 3 lo g_{4} (x)

= 3 lo g_{4} (x) + lo g_{4} (e^{2}) - \frac{1}{2} lo g_{4} (4 x + 1)

Apply log rule:

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

lo g_{4} (e^{2}) = 2 lo g_{4} (e)

= 3 lo g_{4} (x) + 2 lo g_{4} (e) - \frac{1}{2} lo g_{4} (4 x + 1)

ln (x - 9) + ln (x + 9)

\frac{1}{2} ln (\frac{78}{118})

ln (5 \cdot e^{x})

lo g_{10} (x) - 4 lo g_{10} (y) + 9 lo g_{10} (z)

ln (\frac{16}{6})

What is log_{4}((x^3e^2)/((4x+1)^{1/2)}) ?
The solution to log_{4}((x^3e^2)/((4x+1)^{1/2)}) is 3log_{4}(x)+2log_{4}(e)-1/2 log_{4}(4x+1)