![2^{2x+11}=3^{x-31}](/tex.php?f=2^{2x+11}=3^{x-31})
take the logarithm of both sides (the base of the log doesn't matter):
![\ln2^{2x+11}=\ln3^{x-31}](/tex.php?f=\ln2^{2x+11}=\ln3^{x-31})
apply the exponent property - this says that
:
![(2x+11)\ln2=(x-31)\ln3](/tex.php?f=(2x+11)\ln2=(x-31)\ln3)
distribute the log terms on both sides:
![(2\ln2)x+11\ln2=(\ln3)x-31\ln3](/tex.php?f=(2\ln2)x+11\ln2=(\ln3)x-31\ln3)
group like terms together:
![(2\ln2-\ln3)x=-11\ln2-31\ln3](/tex.php?f=(2\ln2-\ln3)x=-11\ln2-31\ln3)
divide through both sides by the coefficient on
:
![x=-\dfrac{11\ln2+31\ln3}{2\ln2-\ln3}](/tex.php?f=x=-\dfrac{11\ln2+31\ln3}{2\ln2-\ln3})
you can do some additional rewriting here using the properties of the logarithm to simplify things if you like:
![x=-\dfrac{\ln2^{11}+\ln3^{31}}{\ln2^2-\ln3}](/tex.php?f=x=-\dfrac{\ln2^{11}+\ln3^{31}}{\ln2^2-\ln3})
![x=-\dfrac{\ln2^{11}3^{31}}{\ln\frac43}](/tex.php?f=x=-\dfrac{\ln2^{11}3^{31}}{\ln\frac43})
![x=-\log_{4/3}2^{11}3^{31}](/tex.php?f=x=-\log_{4/3}2^{11}3^{31})