If log2 m = a and log2 n = b, express each of the following in terms of a and b. a) log4 m²n³
b) log8 m/n²
c) logmn 8n

Let’s start by simplifying log (25/8), i assume 10 to be base of log:

log(25/8)=

log 25 - log 8=

log 5^2 - log 2^3 =

2 log 5 - 3 log 2 =

2 log (10/2) - 3 log 2 =

2 (log 10-log 2) - 3 log 2 =

2(1-log 2) - 3 log 2 =

2 - 2 log 2 - 3 log 2 =

2- 5 log 2 =

Step-by-step explanation:

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the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12