Question
Use properties of logarithms to condense the logarithmic expression below. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.
one fourth ln x plus ln y
Question content area bottom
Part 1
one fourth ln x plus ln y
=
In left parenthesis yx Superscript one fourth Baseline right parenthesis
(Simplify your answer.)
one fourth ln x plus ln y
Question content area bottom
Part 1
one fourth ln x plus ln y
=
In left parenthesis yx Superscript one fourth Baseline right parenthesis
(Simplify your answer.)
Answers
GPT-4o mini
To condense the expression \( \frac{1}{4} \ln x + \ln y \) into a single logarithm whose coefficient is 1:
1. First, apply the power rule of logarithms, which states that \( a \ln b = \ln(b^a) \). Here, we can rewrite \( \frac{1}{4} \ln x \) as:
\[
\ln(x^{1/4})
\]
2. Now we can combine the two logarithmic expressions using the property that \( \ln a + \ln b = \ln(ab) \):
\[
\ln(x^{1/4}) + \ln(y) = \ln(x^{1/4} \cdot y)
\]
3. Therefore, the condensed expression is:
\[
\ln\left(y x^{1/4}\right)
\]
So, the final answer is:
\[
\frac{1}{4} \ln x + \ln y = \ln\left(y x^{1/4}\right)
\]
1. First, apply the power rule of logarithms, which states that \( a \ln b = \ln(b^a) \). Here, we can rewrite \( \frac{1}{4} \ln x \) as:
\[
\ln(x^{1/4})
\]
2. Now we can combine the two logarithmic expressions using the property that \( \ln a + \ln b = \ln(ab) \):
\[
\ln(x^{1/4}) + \ln(y) = \ln(x^{1/4} \cdot y)
\]
3. Therefore, the condensed expression is:
\[
\ln\left(y x^{1/4}\right)
\]
So, the final answer is:
\[
\frac{1}{4} \ln x + \ln y = \ln\left(y x^{1/4}\right)
\]