How do you expand $ln [\frac{5 e 4 {\sqrt{t}}^{5}}{a m}]$ $Ln[(5e4sqrtt^5)/(am)]$ ?

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How do you expand $ln [\frac{5 e 4 {\sqrt{t}}^{5}}{a m}]$ $Ln[(5e4sqrtt^5)/(am)]$ ?

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Answer 1 · 2017-07-06T15:42:31

$ln [\frac{5 e 4 {\sqrt{t}}^{5}}{a m}]$ $ln[frac{5e4sqrtt^5}{am}]$
$= ln 20 + 1 + \frac{5}{2} ln t - ln a - ln m$ $= ln20 + 1 + 5/2lnt - lna - lnm$

Explanation:

$ln [\frac{5 e 4 {\sqrt{t}}^{5}}{a m}]$ $ln[frac{5e4sqrtt^5}{am}]$

Rewrite the numerator:
$= ln [\frac{20 e t^{\frac{5}{2}}}{a m}]$ $= ln[ frac{20et^(5/2)}{am} ]$

Separate numerator and denominator:
$= ln (20 e t^{\frac{5}{2}}) - ln (a m)$ $= ln(20et^(5/2)) - ln(am)$

Expand each natural log using the logarithm rule of products:
$= [ln 20 + ln e + ln (t^{\frac{5}{2}})] - [ln a + ln m]$ $= [ln20 + color(blue)(lne) + ln(t^(5/2))] - [ln a + ln m]$

$Note that ln e = 1$ $color(blue)("Note that "lne=1)$

$= ln 20 + 1 + \frac{5}{2} ln t - ln a - ln m$ $= ln20 + 1 + 5/2lnt - lna - lnm$

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How do you expand $ln [\frac{5 e 4 {\sqrt{t}}^{5}}{a m}]$ $Ln[(5e4sqrtt^5)/(am)]$ ?

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