High School

Part A

Which expression is equivalent to [tex]\log\left(\frac{5x}{7}\right)[/tex]?

1) [tex]\log 5 + \log x - \log 7[/tex]
2) [tex]\log 5 + \log x + \log 7[/tex]
3) [tex]\log 5 \times \log x - \log 7[/tex]
4) [tex]\log 5 \times \log x + \log 7[/tex]

Answer :

The expression log((5x)/(7)) is equivalent to log5 + logx - log7, following the logarithmic properties of products and quotients. The correct option is 1.

The expression log((5x)/(7)) is equivalent to log5 + logx - log7. This is because of the properties of logarithms which state:

  • The logarithm of a product of two numbers is the sum of the logarithms of those two numbers. Therefore, log(5x) can be written as log5 + logx.
  • The logarithm of the number resulting from the division of two numbers is the difference between the logarithms of those two numbers. Hence, log((5x)/7) translates to log5 + logx - log7.

Thus, the correct answer is option 1): log5 + logx - log7.

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