Question

Write the expression as a single logarithm. (Simplify your answer.)2log(6)-2log(3)

Answer 1

Answer:

log 4

Explanation:

Given the logarithm expression:

$2\log (6)-2\log (3)$

To write the expression as a single logarithm, first factor out 2.

$=2\lbrack\log (6)-\log (3)\rbrack$

Next, when logarithms in the same base are being subtracted, we can combine it as follows:

$\begin{gathered} \log (A)-\log (B)=\log (\frac{A}{B})\implies\log (6)-\log (3)=\log (\frac{6}{3}) \\ \implies2\lbrack\log (6)-\log (3)\rbrack=2\log (\frac{6}{3})=2\log (2) \end{gathered}$

Finally, when a number is multiplying a logarithm expression, we can rewrite it as follows:

$\begin{gathered} x\log y\implies\log y^x \\ \implies2\log (2)=\log 2^2=\log 4 \end{gathered}$

Thus, the expression written as a single logarithm gives the equality:

$2\log (6)-2\log (3)=\log 4$