Interactive Example Questions Fill in the blanks Theorem If

Interactive Example Questions Fill in the blanks: Theorem: If b, z, and y are positive numbers, with b1, then log(zy) = log(z) + logb(y) and log -log(z)-log(s). Proof: Suppose b, z, and y are positive numbers, with b1, and let s=log(z) and t=log,(y). Then z = and y=

Solution

s = log_b (x)

==> x = b^s

t = log_b (y)

==> y = b^t

Hence xy = b^s b^t = b^{s + t}        ; since a^m aⁿ = a^m+n

x/y = b^s/b^t = b^{s -t}     ; since a^m/aⁿ = a^m-n

Therefore log_b xy = log_b b^s+t = (s + t) log_b b

==> (s + t) (1)              (since log_a a = 1)

==> s + t

==> log_b x + log_b y

log_b (x/y) = log_b b^{s - t} = (s - t) log_b b

==> (s - t) (1)

==> s - t

==> log_b x - log_b y