Question - Solving Exponential Equation with Substitution
Solution:
The image shows an equation:\[e^{2x} - 4 \cdot 2^x + 4 = 0\]To solve this equation, let's apply a substitution. Notice that the terms $$e^{2x}$$ and $$2^x$$ suggest that we can set $$u = 2^x$$. Then $$e^{2x} = (e^x)^2 = (2^x)^2 = u^2$$. Our equation now becomes:\[u^2 - 4u + 4 = 0\]Now, factor this quadratic equation:\[(u - 2)^2 = 0\]Solving for $$u$$, we get $$u = 2$$.Since we set $$u = 2^x$$, we now have:\[2^x = 2\]Taking the logarithm base 2 of both sides gives:\[x \cdot \log_2(2) = \log_2(2)\]\[\Rightarrow x = 1\]So, the solution to the equation is $$x = 1$$.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com