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Answer :
To expand the logarithm given by [tex]\(\log_4 \sqrt{\frac{3x}{5}}\)[/tex] completely, we will apply the properties of logarithms step-by-step.
1. Square Root Property:
The first step is to deal with the square root. The square root of any expression can be rewritten using a power of [tex]\( \frac{1}{2} \)[/tex]. Therefore:
[tex]\[
\log_4 \sqrt{\frac{3x}{5}} = \log_4 \left(\frac{3x}{5}\right)^{\frac{1}{2}}
\][/tex]
Using the property of logarithms, [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex], we can rewrite this as:
[tex]\[
\frac{1}{2} \cdot \log_4 \left(\frac{3x}{5}\right)
\][/tex]
2. Quotient Property:
Next, apply the quotient property, which says [tex]\(\log_b \left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\)[/tex]. Therefore:
[tex]\[
\frac{1}{2} \cdot \left(\log_4 (3x) - \log_4 5\right)
\][/tex]
3. Product Property:
Now, in the term [tex]\(\log_4 (3x)\)[/tex], use the product property: [tex]\(\log_b(a \cdot c) = \log_b(a) + \log_b(c)\)[/tex]. This gives us:
[tex]\[
\frac{1}{2} \cdot \left(\log_4 3 + \log_4 x - \log_4 5\right)
\][/tex]
4. Distribute the [tex]\(\frac{1}{2}\)[/tex]:
Finally, distribute the [tex]\(\frac{1}{2}\)[/tex] across each term:
[tex]\[
\frac{1}{2} \log_4 3 + \frac{1}{2} \log_4 x - \frac{1}{2} \log_4 5
\][/tex]
So, the completely expanded form of the logarithm [tex]\(\log_4 \sqrt{\frac{3x}{5}}\)[/tex] is:
[tex]\[
\frac{1}{2} \log_4 3 + \frac{1}{2} \log_4 x - \frac{1}{2} \log_4 5
\][/tex]
This matches with the choice provided in the list of options.
1. Square Root Property:
The first step is to deal with the square root. The square root of any expression can be rewritten using a power of [tex]\( \frac{1}{2} \)[/tex]. Therefore:
[tex]\[
\log_4 \sqrt{\frac{3x}{5}} = \log_4 \left(\frac{3x}{5}\right)^{\frac{1}{2}}
\][/tex]
Using the property of logarithms, [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex], we can rewrite this as:
[tex]\[
\frac{1}{2} \cdot \log_4 \left(\frac{3x}{5}\right)
\][/tex]
2. Quotient Property:
Next, apply the quotient property, which says [tex]\(\log_b \left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\)[/tex]. Therefore:
[tex]\[
\frac{1}{2} \cdot \left(\log_4 (3x) - \log_4 5\right)
\][/tex]
3. Product Property:
Now, in the term [tex]\(\log_4 (3x)\)[/tex], use the product property: [tex]\(\log_b(a \cdot c) = \log_b(a) + \log_b(c)\)[/tex]. This gives us:
[tex]\[
\frac{1}{2} \cdot \left(\log_4 3 + \log_4 x - \log_4 5\right)
\][/tex]
4. Distribute the [tex]\(\frac{1}{2}\)[/tex]:
Finally, distribute the [tex]\(\frac{1}{2}\)[/tex] across each term:
[tex]\[
\frac{1}{2} \log_4 3 + \frac{1}{2} \log_4 x - \frac{1}{2} \log_4 5
\][/tex]
So, the completely expanded form of the logarithm [tex]\(\log_4 \sqrt{\frac{3x}{5}}\)[/tex] is:
[tex]\[
\frac{1}{2} \log_4 3 + \frac{1}{2} \log_4 x - \frac{1}{2} \log_4 5
\][/tex]
This matches with the choice provided in the list of options.
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