logarithms and ln

Question

Given log_ba=5 find the following:

a) log_ba³

b)log_b(ab)

c)log_b(b²/a)

d)ln(a)/ln(b)

e)b⁵=a

f)log_bb

Answer 1

log(b)a=5

a) log(b)a³=3*log(b)a=3*5=15

 

b) log(b)ba=log(b)b+log(b)a=1+5=6

 

c) log(b)(b²/a)=log(b)b²-log(b)a=2log(b)b-log(b)a=2*1-5=2-5=-3

 

d) ln(a)/ln(b)=log(e)(a)/log(e)(b)

    log(e)a=log(b)a/log(b)e using rule for change of base

    log(e)b=log(b)b/log(b)e using rule for change of base

    [log(b)a/log(b)e]/[log(b)b/log(b)e]

    [log(b)a/log(b)e]*[log(b)e/log(b)b]

    log(b)a/log(b)b

    5/1

    5

 

e) b⁵=a is the same as log(b)a=5

 

f) log(b)b=1