Question

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

Answer 1

Answer:

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

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

$3.ln(\frac{e^3}{f} ) = 3 ln(e) - ln (f)$  TRUE

Step-by-step explanation:

Here, let us first state the logarithmic rules.

$log (a) - log (b) =log(\frac{a}{b} )\\log (ab) = log a + log b\\log(a^m) = m \times log (a)$

Now, here the given expressions are:

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

The given statement is FALSE as:

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

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

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

The given statement is TRUE .

$3.ln(\frac{e^3}{f} ) = 3 ln(e) - ln (f)$

The given statement is TRUE as:

$ln(\frac{e^3}{f} ) = ln(e^3) - ln (f) = 3 ln(e) - ln (f)$