log(base 5)(1/5) + ln e - log (base 8)(2) + log(0.001) - log(base 8)(16) + ln e^3?

Question

How do I evaluate this using the properties of logarithms and without a calculator??????

Answer 1

break each term up and it becomes much easier to look at

log (base 5) (1/5)=x or
5^x = 1/5 so x=-1

ln e = log(base e) e = x or e^x = e so x=1

log (base 8) (2) = x or
8^x = 2 x=1/3

log (0.001) = x or
10^x = 0.001 so x=-1/3

log (base 8) (16) = x
8^x = 16 so x=4/3 (8^(1/3)=2 then 2^4=16)

ln e^3 = log (base e) e^3 or
e^x = e^3 so x=3 or ln e^3 = 3ln e = 3

put all the terms back in the equation:

-1+1-(1/3)+(1/3)-(4/3)+3= 5/3

hope this helps you.

Answer 2

Logarithms mean what number would you need to use as an exponent on a base number to equal the given number. That sounds complicated, but it isn't. What number would you need to use as an exponent of 2 to equal 8? You can see that 2^3 is 8, therefore the answer is 3. Your teacher is making you really learn with this problem since most calculators will not perform these calculations, but the answers are very easy.

Break it down into parts.

log(base 5)(1/5)...... means what power number would you need to raise 5 to equal 1/5th. The answer is -1.

ln(e)...... means what power number would you need to raise (e) to equal (e). The answer is 1.

log(base 8)(2)...... means what power number would you need to raise 8 to equal 2. The answer is 1/3rd.

log(0.001)...... means what power number would you need to raise 10 to equal 0.001. The answer is -3.

log(base 8)(16)...... means what power number would you need to raise 8 to equal 16. NOW this one gets tricky.....
You know that 16 can be factored into 2*8. The logarithms of factors are additive. Now you know that log(base 8)(2)=1/3rd because you did that above. And the log(base 8)(8) is 1. Since logs of factors are additive the log(base 8)(16)=1/3rd+1=1&1/3.

ln(e^3)...... means what power number would you need to raise (e) to equal (e)^3. The answer is 3.

Now do the arithmetic to add/subtract up all your answers....
-1+1-1/3+(-3)-(1+1/3)+3
= -1+1-1/3-3-1-1/3+3
= -1 & 2/3

Answer 3

= -1 +1 -log(8) 8^1/3+log10^-3 -log(8)(2*8)=
-1/3 -3 -1/3 -1 +3 because log(8)(2*8)=log (8)2+log(8) 8 So the result is -1-2/3 = -5/3

Answer 4

= log(base 5)(5^-1) + log(base e)(e) - log(base 2^3)(2) + log(base 10)(10^-3) - log(base 2^3)(2^4) + log(base e)(e^3)

= -1 + 1 -(1/3) +(-3) - (4/3) + 3

= -5/3 ANS