Properties of Logs?

Question

Suppose that In 2 = a and In 3 = b. Use properties of logs to write each log in terms of a and b.

QUESTIONS:

(1) In(2/3)

(2) In(3/e)

(3) In 0.5

(4) In {fourth root of 48}

NOTE:

In means natural log.

Answer 1

1. ln(2/3) = ln2 - ln3 = a-b

2. ln(3/e) = ln3 - ln(e) = b-1 --------- ln(e) = 1

3. ln(0.5) = ln(1/2) = -a

4. In {fourth root of 48}
= (1/4) ln(48)
= 1/4 ln(2^4 * 3)
= 1/4 (4ln2 + ln3)
= 1/4(4a + b)
= a + b/4

Answer 2

Hey there!

Here's the first answer.

ln(2/3) --> Write the problem.
ln 2-ln 3 --> Recall the quotient property of logarithms, ln a/b= ln a-ln b.
a-b Substitute a for ln 2 and b for ln 3.

So the answer is a-b.

Here's the second answer.

ln(3/e) --> Write the problem.
ln 3-ln e --> Recall the quotient property of logarithms, ln a/b= ln a-ln b.
ln 3-1 --> Recall that ln e is 1.
b-1 Substitute b for ln 3.

So the answer is b-1.

Here's the third answer.

ln 0.5 --> Write the problem.
ln 1/2 --> Rewrite 0.5 as a fraction, which is 1/2.
ln 1-ln 2 --> Recall the quotient property of logarithms, ln a/b= ln a-ln b.
0-ln 2 --> Recall that ln 1 is 0.
-ln 2 --> Simplify the above expression.
-a Substitute a for ln 2.

So the answer is -a.

Here's the last answer.

ln(48^1/4) --> Write the problem.
1/4ln(48) --> Recall the power property of logarithms, ln m^n=n*ln m.
1/4ln(16*3) --> Change 48 as the product of 16 and 3.
1/4(ln(16)+ln(3)) --> Recall the product property of logarithms, ln mn=ln m+ln n.
1/4(ln(2^4)+ln(3)) --> Change 16 as 2^4.
1/4(4ln(2)+ln(3)) --> Recall the power property of logarithms, ln m^n=n*ln m.
ln(2)+1/4ln(3) --> Use the distributive property, a(b+c)=ab+ac.
a+1/4b --> Substitute a for ln 2 and b for ln 3.
a+b/4 Rewrite 1/4b as b/4.

So the answer is a+b/4. If you want, you can rewrite it as (4a+b)/4.

Hope it helps!

Answer 3

1) ln(2/3) = ln2 - ln3 = a - b

2) ln (3/a) = ln3 - ln e = ln3 - 1 = b-1

3) ln (0.5) = ln(1/2) = ln 1 - ln 2 = 0 - ln 2 = -a

4) ln (48^0.25) = 0.25 * ln(48) = 0.25 * ln (3 * 16) =
= 0.25 * [ln(3) + ln(16)] = 0.25 * [ln 3 + ln(2^4)] =
= 0.25 * (ln 3 + 4ln2) = 0.25 (b + 4a) = 0.25b + a

Answer 4

1. a-b
2. b-1
3. =-ln(2)= -a
4. =ln(48)/4 = (4*a+b)/4

Answer 5

1) a - b
2) b - 1
3) -a
4) (1/4)(4*a + b)