How do you simplify ln 3 − 2 ln 8 + ln 16ln3−2ln8+ln16?

2 Answers
Alex C.
Mar 25, 2018

ln(3/4)ln(34)

Explanation:

We have to use the Logarithm Properties.

ln(3)-2ln(8)+ln(16)ln(3)−2ln(8)+ln(16)

We can rewrite the initial expression using the Power rule in this way
ln(3)-ln(8^2)+ln(16)ln(3)−ln(82)+ln(16)

ln(3)-ln(64)+ln(16)ln(3)−ln(64)+ln(16)

Here we use the Quotient rule
ln(3/64)+ln(16)ln(364)+ln(16)

And here the Product rule
ln(3/64*16)ln(364⋅16)

ln(48/64)ln(4864)

Finally we get the semplified version :
ln(3/4)ln(34)

Jim G.
Mar 25, 2018

ln(3/4)ln(34)

Explanation:

"using the "color(blue)"laws of logarithms"using the laws of logarithms

•color(white)(x)logx^nhArrnlogx∙xlogxn⇔nlogx

•color(white)(x)logx+logyhArrlog(xy)∙xlogx+logy⇔log(xy)

•color(white)(x)logx-logyhArrlog(x/y)∙xlogx−logy⇔log(xy)

rArrln3-ln8^2+ln16⇒ln3−ln82+ln16

=ln((3xx16)/64)=ln(48/64)=ln(3/4)=ln(3×1664)=ln(4864)=ln(34)

Related questions
See all questions in Natural Logs
Impact of this question
9619 views around the world
You can reuse this answer
Creative Commons License