What is ln(1/A e^{x+6}) ?

The solution to ln(1/A e^{x+6}) is x+6-ln(A)

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simplify ln(1/A e^{x+6})

simplify

ln (\frac{1}{A} e^{x + 6})

x + 6 - ln (A)

Solution steps

ln (\frac{1}{A} e^{x + 6})

Apply log rule:

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

ln (\frac{1}{A} e^{x + 6}) = ln (\frac{1}{A}) + ln (e^{x + 6})

= ln (\frac{1}{A}) + ln (e^{x + 6})

Apply log rule:

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

ln (e^{x + 6}) = (x + 6) ln (e)

= ln (\frac{1}{A}) + (x + 6) ln (e)

Apply log rule:

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

ln (\frac{1}{A}) = ln (1) - ln (A)

= ln (1) - ln (A) + ln (e) (x + 6)

ln (1) = 0

(x + 6) ln (e) = x + 6

= 0 - ln (A) + x + 6

Group like terms

= x - ln (A) + 0 + 6

Add the numbers:

0 + 6 = 6

= x + 6 - ln (A)

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