Chapter 5 - 5.4 - Exponential and Logarithmic Equations - 5.4 Exercises - Page 395: 39

$x=0$ $x=0.861$

$4^x=5^{x^2}$ $\log_44^x=\log_45^{x^2}$ $x=x^2\log_45$ $-x^2\log_45+x=0$ $-x(x\log_45-1)=0$ $-x=0$ $x=0$ $x\log_45-1=0$ $x\log_45=1$ $x=\frac{1}{\log_45}=\frac{1}{\frac{\ln5}{\ln4}}=\frac{\ln4}{\ln5}=0.861$