color(white)(xxx)2^x=3^(x-1)
=>log_3 2^x=x-1
The logarithm of the x^(th) power of a number is x times the logarithm of the number itself:
color(white)(xxx)xlog_3 2=x-1
Multiply both sides by color(red)(1/x)
color(white)(xxx)color(red)(1/x*)xlog_3 2=color(red)(1/x*)(x-1)
=>(x-1)/x=log_3 2
Add color(red)(-1) to both sides:
color(white)(xxx)(x-1)/xcolor(red)(-1)=log_3 2color(red)(-1)
=>1/x=log_3 2-1
=>1/x=log_3 2-color(blue)(log_3 3)color(white)(xxxxx) (because for AAainRR, a^1=a)
The logarithm of the ratio of two numbers is the difference of the logarithms:
=>1/x=log_3 (2/3)
=>x=log_(2/3)3
