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

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

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

simplify

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

- 4 x - 2 ln (2)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

ln (1) = 0

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

= 0 - (4 x + 2 ln (2))

0 - (2 ln (2) + 4 x) = - (2 ln (2) + 4 x)

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

= - 4 x - 2 ln (2)

- \frac{1}{5} ln (3)

lo g_{5} (x^{3} y^{6})

lo g_{6} (- 2 lo g_{10} (y))

ln (\frac{x}{y - z})

4 ln (\frac{36}{35}) - 0.05