Expand the expression log(8x/5) using the properties of logarithms

Question

1)log8+logx
2)log8+logx-log5
3)8logx-log5
4)log8/log5 x logx/log5

Answer 1

To expand the expression \(\log\left(\frac{8x}{5}\right)\) using the properties of logarithms, we can use the following properties:

1. \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\) (the quotient rule)
2. \(\log(ab) = \log(a) + \log(b)\) (the product rule)

Applying these properties to \(\log\left(\frac{8x}{5}\right)\):

\[ \log\left(\frac{8x}{5}\right) = \log(8x) - \log(5) \]

Next, we can apply the product rule to \(\log(8x)\):

\[ \log(8x) = \log(8) + \log(x) \]

Putting it all together, we have:

\[ \log\left(\frac{8x}{5}\right) = \log(8) + \log(x) - \log(5) \]

Thus, the expanded expression is:

\[ \log(8) + \log(x) - \log(5) \]

This corresponds to option 2) \(\log 8 + \log x - \log 5\).