Questions: Simplify the logarithm 4 log5 x + 3 log5 y - 2 log5 z

Solution Steps

To simplify the given logarithmic expression, we can use the properties of logarithms. Specifically, we can use the power rule, which states that $a \log_b c = \log_b (c^a)$, and the product and quotient rules, which state that $\log_b m + \log_b n = \log_b (mn)$ and $\log_b m - \log_b n = \log_b \left(\frac{m}{n}\right)$, respectively. Applying these rules will allow us to combine the terms into a single logarithm.

The given expression is $4 \log_{5} x + 3 \log_{5} y - 2 \log_{5} z$. Using the power rule of logarithms, which states that $a \log_b c = \log_b (c^a)$, we can rewrite each term as:

• $4 \log_{5} x = \log_{5} (x^4)$
• $3 \log_{5} y = \log_{5} (y^3)$
• $-2 \log_{5} z = \log_{5} (z^{-2})$

Next, we use the product rule $\log_b m + \log_b n = \log_b (mn)$ and the quotient rule $\log_b m - \log_b n = \log_b \left(\frac{m}{n}\right)$ to combine the terms: $$\log_{5} (x^4) + \log_{5} (y^3) - \log_{5} (z^{-2}) = \log_{5} \left(\frac{x^4 \cdot y^3}{z^2}\right)$$

Final Answer