Show that each equation is true for c 0 and c notequalto 1

Show that each equation is true for c > 0 and c notequalto 1. log, . 48 - (log_c 3 + log_c 2) = log_c 8 7 log_c 4 = 14 log_c 2 1/2 (log_C 2 + log_C 6) = log_c 2 + log_c Squareroot 3 log_c(5 c)^2 = 2 (log_c 5 + 1)

Solution

a) log_c48 - (log_c3 + log_c2) = log_c8

Starting with left side

log_c48 - (log_c2 + log_c3)

We have to use the following log rules

log_ab + log_ac=log_abc

log_ab - log_ac = log_a(b/c)

log_c48 - log_c(2*3)  

=   log_c48 - log_c6= log_c(48/6)=log_c8 that is equal to right side

b) 7log_c4= 14 log_c2

Starting with left side

7log_c4

We have to use the following log rule

log_ab^c= c log_ab

7log_c2² = 7*2 log_c2 = 14log_c2 that is equal to right side

c)1/2(log_c2 + log_c6)= log_c2 + log_csqrt3

Starting with left side

1/2(log_c2 + log_c6)

1/2(log_c2 + log_c(2*3))

1/2(log_c2 + log_c2 +log_c3)

1/2(2log_c2 + log_c3)   = log_c2 + 1/2log_c3 = log_c2 + log_csqrt3   that is equal to right side

d) log_c(5c)² = 2(log_c5 + 1)

Starting with the left side

log_c(5c)²   = 2log_c5c

               = 2(log_c5 + log_cc)

and log_cc=1

therefore log_c(5c)²= 2(log_c5 + 1) that is the right side