if ln(a)=2, ln(b)=3 and ln(c)=5, evaluate the follwing:

Question

ln[(a^4)/(b^-2*c^-1)]

 

ln[(sqrt)(b^-4)(c^-3)(a^3)]

 

ln(a^-3*b^-3)/ln((bc)^4)

 

ln(c^3)* ln(a/b^4)^-3

Answer 1

The following log rules are needed here:

1) ln(x*y) = ln x + ln y

2) ln(x/z) = ln x - ln z

3) ln(x^r) = r ln x

 

Your first problem:

 

ln[(a⁴)/(b^-2*c^-1)]

Use rule 2 to split up the numerator and denominator

= ln(a⁴) - ln(b^-2*c^-1)

Use rule 1 to split up the 2nd log, then distribute the - sign

= ln(a⁴) - [ln(b^-2) + ln(c^-1)]

= ln(a⁴) - ln(b^-2) - ln(c^-1)

Now use rule 3 to move the powers in front

= 4 ln a - (-2) ln b - (-1) ln c

= 4 ln a + 2 ln b + ln c

Plug in the values given,

= 4(2) + 2(3) + (5)

= 19

 

For the second problem, remember that roots can be rewritten as fractional exponents. In particular √(xyz) = (xyz)^1/2

 

For the third problem, don't confuse ln(x)/ln(y) with rule 2. You cannot expand ln(x)/ln(y), only if the division were inside the log.

 

For the fourth problem, the same is true with the multiplication between logs. You cannot expand that, the product would have to be inside the log to use rule 1.

 

Compare the first solution to the other problems, and use the hints given, and you should be able to solve the rest. Please comment after you've tried if you still need help.