Answer:
[tex]1. log_3(cd)^4 = 4 log_3(c )+ log_3(d)[/tex] FALSE
[tex]2. \frac{2}{3} (ln(a) +ln(b)) = ln(\sqrt[3]{a^2b^2} )[/tex] TRUE
[tex]3.ln(\frac{e^3}{f} ) = 3 ln(e) - ln (f)[/tex] TRUE
Step-by-step explanation:
Here, let us first state the logarithmic rules.
[tex]log (a) - log (b) =log(\frac{a}{b} )\\log (ab) = log a + log b\\log(a^m) = m \times log (a)[/tex]
Now, here the given expressions are:
[tex]1. log_3(cd)^4 = 4 log_3(c )+ log_3(d)[/tex]
The given statement is FALSE as:
[tex]log_3(cd)^4 = 4 log_3(cd) = 4log_3(c) + 4log_3(d)[/tex]
[tex]2. \frac{2}{3} (ln(a) +ln(b)) = ln(\sqrt[3]{a^2b^2} )[/tex]
[tex]\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} )[/tex]
The given statement is TRUE .
[tex]3.ln(\frac{e^3}{f} ) = 3 ln(e) - ln (f)[/tex]
The given statement is TRUE as:
[tex]ln(\frac{e^3}{f} ) = ln(e^3) - ln (f) = 3 ln(e) - ln (f)[/tex]
