Problem

Condense the following expression by using the properties of logarithms. \[ 4 \log x+6 \log y-10 \log z^{2} \]

Solution

Step 1 :We are given the expression \(4 \log x+6 \log y-10 \log z^{2}\).

Step 2 :We can use the properties of logarithms to simplify this expression.

Step 3 :First, we use the property \(\log a^n = n \log a\) to rewrite the expression as \(\log x^{4} + \log y^{6} - \log (z^{2})^{10}\).

Step 4 :Next, we use the properties \(\log a + \log b = \log (ab)\) and \(\log a - \log b = \log \left(\frac{a}{b}\right)\) to combine the logs. This gives us \(\log \left(\frac{x^{4} y^{6}}{(z^{2})^{10}}\right)\).

Step 5 :Simplifying the denominator gives us \(\log \left(\frac{x^{4} y^{6}}{z^{20}}\right)\).

Step 6 :Final Answer: \(\boxed{\log \left(\frac{x^{4} y^{6}}{z^{20}}\right)}\)

From Solvely APP
Source: https://solvelyapp.com/problems/8539/

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