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Chapter 3: Problem 56

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$7 \ln x-3 \ln y$$

Short Answer

Expert verified
The given expression can be simplified to \(\ln \left(\frac{x^7}{y^3}\right)\).

Step by step solution

01

Apply Logarithmic Properties

Starting with the given expression, \(7 \ln x - 3 \ln y\), apply the rule \(a \log_b c = \log_b (c^a)\) to get \(\ln (x^7) - \ln (y^3)\).
02

Further Simplification

Then, apply the rule \(\log_b(ac) = \log_b a + \log_b c\) and invert it as subtracting logarithms is equivalent to division into one logarithm, resulting to \(\ln \left(\frac{x^7}{y^3}\right)\).

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