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

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

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

simplify

ln (\frac{e ^{2 x} ( x + 4 ) ^{2}}{2 x + 1})

2 x + 2 ln (x + 4) - ln (2 x + 1)

Solution steps

ln (\frac{e ^{2 x} ( x + 4 ) ^{2}}{2 x + 1})

Apply log rule:

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

ln (\frac{e ^{2 x} ( x + 4 ) ^{2}}{2 x + 1}) = ln (e^{2 x} (x + 4)^{2}) - ln (2 x + 1)

= ln (e^{2 x} (x + 4)^{2}) - ln (2 x + 1)

Apply log rule:

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

ln (e^{2 x} (x + 4)^{2}) = ln (e^{2 x}) + ln ((x + 4)^{2})

= ln (e^{2 x}) + ln ((x + 4)^{2}) - ln (2 x + 1)

Apply log rule:

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

ln (e^{2 x}) = 2 x ln (e)

= 2 x ln (e) + ln ((x + 4)^{2}) - ln (2 x + 1)

Apply log rule:

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

ln ((x + 4)^{2}) = 2 ln (x + 4)

= 2 x ln (e) + 2 ln (x + 4) - ln (2 x + 1)

2 x ln (e) = 2 x

= 2 x + 2 ln (x + 4) - ln (2 x + 1)

lo g_{9} (81 x^{8})

lo g_{5} (\frac{125}{x + 7})

lo g_{b} (108 b)

5 lo g_{10} (2) + lo g_{10} (3)

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