Question #018b8

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Answer 1 · 2017-09-04T15:06:52

#ln(131072/9)#

We're asked to simplify the expression

#8ln2 + 3ln8 - 2ln3#

We can use one of the properties of logarithms to help us:

#ul(alnb = ln(b^a)#

For the first term:

#color(red)(8)lncolor(green)(2) = ln(color(green)(2)^color(red)(8))#

And similarly for the other two terms:

#= ln(color(green)(2)^color(red)(8)) + ln(color(green)(8)^color(red)(3)) - ln(color(green)(3)^color(red)(2))#

Now, we can use another property:

#ul(lna + lnb = ln(ab)#

and

#ul(lna - lnb = ln(a/b)#

And so we have

#= ln((2^8 * 8^3)/(3^2))#

#= color(blue)(ulbar(|stackrel(" ")(" "ln(131072/9)" ")|)#