Question
Expand the expression log(8x/5) using the properties of logarithms.(1 point) Responses log8⋅logxlog5 Start Fraction logarithm 8 times logarithm x over logarithm 5 End Fraction 8logx−log5 8 logarithm x minus logarithm 5 log8+logx logarithm 8 plus logarithm x log8+logx−log5
Answers
Answered by
GPT-4o mini
To expand the expression \(\log\left(\frac{8x}{5}\right)\) using the properties of logarithms, we can apply the quotient rule and the product rule of logarithms.
The key properties we will use are:
1. \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\) (Quotient Rule)
2. \(\log(ab) = \log(a) + \log(b)\) (Product Rule)
Starting with the expression:
\[
\log\left(\frac{8x}{5}\right)
\]
Using the quotient rule, we get:
\[
\log(8x) - \log(5)
\]
Now we can apply the product rule to \(\log(8x)\):
\[
\log(8) + \log(x) - \log(5)
\]
Thus, the final expanded expression is:
\[
\log(8) + \log(x) - \log(5)
\]
From the options you provided, the correct answer that matches this result is:
\(\log(8) + \log(x) - \log(5)\) (the last option).
The key properties we will use are:
1. \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\) (Quotient Rule)
2. \(\log(ab) = \log(a) + \log(b)\) (Product Rule)
Starting with the expression:
\[
\log\left(\frac{8x}{5}\right)
\]
Using the quotient rule, we get:
\[
\log(8x) - \log(5)
\]
Now we can apply the product rule to \(\log(8x)\):
\[
\log(8) + \log(x) - \log(5)
\]
Thus, the final expanded expression is:
\[
\log(8) + \log(x) - \log(5)
\]
From the options you provided, the correct answer that matches this result is:
\(\log(8) + \log(x) - \log(5)\) (the last option).