To expand the expression \(\log\left(\frac{a^5}{b}\right)\) using the properties of logarithms, we can apply the following rules:
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The logarithm of a quotient: \[ \log\left(\frac{x}{y}\right) = \log(x) - \log(y) \]
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The logarithm of a power: \[ \log(x^n) = n \cdot \log(x) \]
Now, let's apply these rules step-by-step to the given expression.
Starting with: \[ \log\left(\frac{a^5}{b}\right) \]
We can first apply the quotient rule: \[ \log\left(\frac{a^5}{b}\right) = \log(a^5) - \log(b) \]
Next, we apply the power rule to \(\log(a^5)\): \[ \log(a^5) = 5 \cdot \log(a) \]
Now we can substitute this into our expression: \[ \log\left(\frac{a^5}{b}\right) = 5 \cdot \log(a) - \log(b) \]
Therefore, the expanded expression is: \[ \log\left(\frac{a^5}{b}\right) = 5 \log(a) - \log(b) \]