What is logarithm method?????Explain &Solve the following equation using logarithm method:----

Solution:
To use the logarithm method, we have to reduce the expression on each side to one term.

$$\begin{gathered} {25}^{n-1}+100=5^{2n-1} \\ \Rightarrow 5^{2n-1}-{25}^{n-1}=100 \\ \Rightarrow 5^{2n-1}-5^{2\left(n-1\right)}=100 \\ \Rightarrow 5^{2n-1}-5^{2n-2}=100 \\ \Rightarrow 5^{2n-2}\left(5-1\right)=100 \\ \Rightarrow 5^{2n-2}=25 \\ \mathrm{Taking} \mathrm{logarithm} \mathrm{on} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ \log \left(5^{2n-2}\right)=\log \left(25\right) \\ \Rightarrow \log \left(5^{2n-2}\right)=\log \left(5^2\right) \\ \Rightarrow \left(2n-2\right)\log 5=2\log 5 \left(\because \log a^m=m\log a\right) \\ \Rightarrow \left(2n-2\right)=2 \\ \Rightarrow 2n=4 \\ \Rightarrow n=2 \end{gathered}$$

Hence, the value of n is 2.