Mitchell Fowler
06/23/2023 · Elementary School
Use properties of logarithms to condel possible. \( 4 \ln (x+5)-7 \ln x \) \( 4 \ln (x+5)-7 \ln x=\square \)
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Tutor-Verified Answer
Step-by-step Solution
To condense the expression \( 4 \ln (x+5) - 7 \ln x \) using properties of logarithms, we can apply the following properties:
1. \( a \ln b = \ln(b^a) \)
2. \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \)
Let's apply these properties step by step:
1. Use the first property to rewrite \( 4 \ln (x+5) \) and \( -7 \ln x \):
\[
4 \ln (x+5) = \ln((x+5)^4)
\]
\[
-7 \ln x = \ln(x^{-7})
\]
2. Now substitute these back into the expression:
\[
4 \ln (x+5) - 7 \ln x = \ln((x+5)^4) + \ln(x^{-7})
\]
3. Use the second property to combine the logarithms:
\[
\ln((x+5)^4) + \ln(x^{-7}) = \ln\left((x+5)^4 \cdot x^{-7}\right)
\]
4. This simplifies to:
\[
\ln\left(\frac{(x+5)^4}{x^7}\right)
\]
Thus, the condensed form of the expression is:
\[
4 \ln (x+5) - 7 \ln x = \ln\left(\frac{(x+5)^4}{x^7}\right)
\]
So, the final answer is:
\[
\ln\left(\frac{(x+5)^4}{x^7}\right)
\]
Quick Answer
\[
\ln\left(\frac{(x+5)^4}{x^7}\right)
\]
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