<s>[INST] Simplifying $\log_x 25 - \left(\frac{\log_5 x}{4} - 2\log_5 x\right)$ My teacher gave me the following expression to change into a single expression:
$$\log_x 25 - \left(\frac{\log_5 x}{4} - 2\log_5 x\right)$$
This is what I got after working it out:
$$\log_x 25 - \log_5 \left(\frac{x}{4x^2}\right)$$
 [/INST] $$\begin{align}
& \log_{x}25 - \left(\dfrac{\log_{5}x}{4} - 2\log_{5}x\right) \\
= & \log_{x}5^2 + \dfrac{7}{4}\log_{5}x \\
= & 2\log_{x}5 + \dfrac{7}{4}\log_{5}x \\
= & \dfrac{2}{\log_{5}x} + \dfrac{7}{4}\log_{5}x \\
= & \dfrac{7\log_{5}x + 8}{4\log_{5}x} \\
= & \dfrac{\log_{5}x^7 + \log_{5}5^8}{\log_{5}x^4} \\
= & \dfrac{\log_{5}(5^8 \cdot x^7)}{\log_{5}x^4} \\
= & \log_{x^4}(5^8 \cdot x^7) \\
\end{align}$$
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