color(white)(xxx)2^x=3^(x-1)×x2x=3x−1
=>log_3 2^x=x-1⇒log32x=x−1
The logarithm of the x^(th)xth power of a number is xx times the logarithm of the number itself:
color(white)(xxx)xlog_3 2=x-1×xxlog32=x−1
Multiply both sides by color(red)(1/x)1x
color(white)(xxx)color(red)(1/x*)xlog_3 2=color(red)(1/x*)(x-1)×x1x⋅xlog32=1x⋅(x−1)
=>(x-1)/x=log_3 2⇒x−1x=log32
Add color(red)(-1)−1 to both sides:
color(white)(xxx)(x-1)/xcolor(red)(-1)=log_3 2color(red)(-1)×xx−1x−1=log32−1
=>1/x=log_3 2-1⇒1x=log32−1
=>1/x=log_3 2-color(blue)(log_3 3)color(white)(xxxxx)⇒1x=log32−log33××x (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
