Question

How can the logarithmic expression be rewritten? Select True or False for each statement.

Answer 1

`1. log_3(cd)^4  = 4 log_3(c )+ log_3(d)` FALSE

`2. (2)/(3) (ln(a) +ln(b)) =  ln(\sqrt[3]{a^2b^2} )` TRUE

`3.ln((e^3)/(f) )  = 3 ln(e) - ln (f)` TRUE

Explanation:

Here, let us first state the logarithmic rules.

`log (a) - log (b)  =log((a)/(b) )\\log (ab)   = log a + log b\\log(a^m) = m * log (a)`

Now, here the given expressions are:

`1. log_3(cd)^4  = 4 log_3(c )+ log_3(d)`

The given statement is FALSE as:

`log_3(cd)^4   = 4 log_3(cd)  = 4log_3(c) + 4log_3(d)`

`2. (2)/(3) (ln(a) +ln(b)) =  ln(\sqrt[3]{a^2b^2} )`

`(2)/(3) (ln(a) +ln(b))  =  (2)/(3) ln(ab)   =  ln(ab) ^(2)/(3)  \\= ln(\sqrt[3]{a^2b^2} )\\\implies(2)/(3) (ln(a) +ln(b)) =  ln(\sqrt[3]{a^2b^2} )`

The given statement is TRUE .

`3.ln((e^3)/(f) )  = 3 ln(e) - ln (f)`

The given statement is TRUE as:

`ln((e^3)/(f) )  =   ln(e^3) - ln (f)   =  3 ln(e) - ln (f)`