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

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

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simplify ln((2x)/(e^{2x)})

simplify

ln (\frac{2 x}{e ^{2 x}})

ln (2) + ln (x) - 2 x

Solution steps

ln (\frac{2 x}{e ^{2 x}})

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (2 x) - 2 x ln (e)

Apply log rule:

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

ln (2 x) = ln (2) + ln (x)

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

2 x ln (e) = 2 x

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

\frac{1}{2} ln (x - 3) + \frac{1}{2} ln (x + 3) - ln (x)

4.74 + lo g_{10} (\frac{50}{10})

\frac{( ln ∣ x + 4 ∣ - 6 ln ∣ x - 1 ∣ )}{5}

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