21. e^-ln2
25.e^2*e^-2
14.-ln1/2
18. ln8+ln4
19.ln8-2ln2
27.ln 1/2- ln 1/4
24.ln(e^ln2)
25.e^2*e^-2
14.-ln1/2
18. ln8+ln4
19.ln8-2ln2
27.ln 1/2- ln 1/4
24.ln(e^ln2)
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21. e^(-ln2)
First off, anything to the negative power can be 1/(that thing to the positive power)
For example, 2^(-3) is 1/(2^3)
So in this case its 1/(e^(ln2))
e's and ln's cancel each other out, so now you have
1/2.
25. e^2*e^-2
Same here. e^(-2) is actually 1/e^(2). Now we have e^2/e^2, which is 1.
14. Not sure if its -ln (1)/2 or -ln .5. If the question is -ln(1)/2, then....
the natural log of one is zero. 0/2 = 0
18. ln8 + ln4
ln(a) + ln(b) = ln(ab)
In order words, you can multiply 8 and 4 to make 32 and take the natural log of that.
ln 32.
19. ln8 -2ln2
if there is a coefficient (-2 in -2ln2) , you can bring that to the front of the natural log and make it
ln (2^(-2)), or ln (.25)
ln 8 + ln .25
You can multiply the numbers inside the ln's. In this case,
8*.25 = 2
ln 2
27. ln .5 - ln .25
When it comes to subtraction yu can divide the numbers inside the ln's. In this case, its .5/.25, or
.5*4 (2)
ln 2
24.ln(e^ln2)
the natural log of e cancels each other out, leaving the ln 2 at the end left.
ln 2
Hope this clears thing up
First off, anything to the negative power can be 1/(that thing to the positive power)
For example, 2^(-3) is 1/(2^3)
So in this case its 1/(e^(ln2))
e's and ln's cancel each other out, so now you have
1/2.
25. e^2*e^-2
Same here. e^(-2) is actually 1/e^(2). Now we have e^2/e^2, which is 1.
14. Not sure if its -ln (1)/2 or -ln .5. If the question is -ln(1)/2, then....
the natural log of one is zero. 0/2 = 0
18. ln8 + ln4
ln(a) + ln(b) = ln(ab)
In order words, you can multiply 8 and 4 to make 32 and take the natural log of that.
ln 32.
19. ln8 -2ln2
if there is a coefficient (-2 in -2ln2) , you can bring that to the front of the natural log and make it
ln (2^(-2)), or ln (.25)
ln 8 + ln .25
You can multiply the numbers inside the ln's. In this case,
8*.25 = 2
ln 2
27. ln .5 - ln .25
When it comes to subtraction yu can divide the numbers inside the ln's. In this case, its .5/.25, or
.5*4 (2)
ln 2
24.ln(e^ln2)
the natural log of e cancels each other out, leaving the ln 2 at the end left.
ln 2
Hope this clears thing up
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Hi Mary,
You need to do these without a calculator, yes?
Here are a few:
e^2*e^-2
whenever you multiply any similar value, be it e, x, 5, or anything, you can add the exponents, so that becomes e^0 and anything to the zero power is 1.
-ln(x) = ln(1/x) so
e^-ln2 = e^ln(1/2), e^ln is just 1, so 1 * 1/2 = 1/2
by the same rule, -ln (1/2) = ln(1/(1/2) = ln(2)
lnx + lny = ln(x*y) so
ln8 + ln4 = ln(32)
ln(e^ln2) again ln and e sort of cancel each other out, so it's just ln2
ln x - ln y = ln(x/y) so ln (1/2) - ln(1/4) = ln(1/2 / 1/4) = ln 2
you can probably figure out the rest from there. i have to go. good luck.
You need to do these without a calculator, yes?
Here are a few:
e^2*e^-2
whenever you multiply any similar value, be it e, x, 5, or anything, you can add the exponents, so that becomes e^0 and anything to the zero power is 1.
-ln(x) = ln(1/x) so
e^-ln2 = e^ln(1/2), e^ln is just 1, so 1 * 1/2 = 1/2
by the same rule, -ln (1/2) = ln(1/(1/2) = ln(2)
lnx + lny = ln(x*y) so
ln8 + ln4 = ln(32)
ln(e^ln2) again ln and e sort of cancel each other out, so it's just ln2
ln x - ln y = ln(x/y) so ln (1/2) - ln(1/4) = ln(1/2 / 1/4) = ln 2
you can probably figure out the rest from there. i have to go. good luck.
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21. 2
25. e
14. ln2 - ln1 = .69314718
18. ln8*4 = ln32 = 3.465735903
19. ln8 - ln4 = ln8/4 = ln2 = .69314718
27. ln2 = .69314718
24. ln2 = .69314718
25. e
14. ln2 - ln1 = .69314718
18. ln8*4 = ln32 = 3.465735903
19. ln8 - ln4 = ln8/4 = ln2 = .69314718
27. ln2 = .69314718
24. ln2 = .69314718
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21. e^-ln2 = 1/2
25.e^2*e^-2 = 1
14.-ln1/2 = ln2
18. ln8+ln4 = 5*ln2
19.ln8-2ln2 = ln2
27. ln1/2 - ln1/4 = ln2
24.ln(e^ln2) = ln2
25.e^2*e^-2 = 1
14.-ln1/2 = ln2
18. ln8+ln4 = 5*ln2
19.ln8-2ln2 = ln2
27. ln1/2 - ln1/4 = ln2
24.ln(e^ln2) = ln2