\(\log_b{(xy)}=\log_b{(x)}+\log_b{(y)}\)
\(\log_b{\left( \dfrac{x}{y} \right)}=\log_b{(x)}-\log_b{(y)}\)
\(\log_b{(x^n)}=n\log_b{(x)}\)
\(\log_b{(x})=\dfrac{\log_k{(x)}}{\log_k{(b)}}\) (for any base \(k\))
\(\log_b{(1)}=0\) (since \(b^0=1\))
\(\log_b{(b)}=1\) (since \(b^1=b\))
Generally,
If \(f:x\rightarrow a^x\), then \(f^{-1}:x\rightarrow \log_ax\).
Thus,
\(y=\log_ax\) is the inverse of \(a^y=x\).
Solve the equation \(3^{x-4}=50^{x-3}\).
\(\begin{aligned} 3^{x-4}&=50^{x-3} \\ (x-4)\log3&=(x-3)\log50 \\ x\log3-4\log3&=x\log50-3\log50 \\ x\log3-x\log50&=-3\log50+4\log3 \\ x(\log3-\log50)&=-3\log50+4\log3 \\ x&=\dfrac{-3\log50+4\log3}{\log3-\log50} \\ &=2.610 .\end{aligned}\)
Thus, \(x=2.610\) is the solution for the equation.
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