Answer
$x=7$
Work Step by Step
Recall the quotient property of a logarithm (pg. 462):
$\log_b{\frac{m}{n}}=\log_b{m}-\log_b{n}$
Also, recall the definition of a logarithm (pg. 451):
$\log_{b}{x}=y$ iff $b^y=x$
We are asked to solve:
$\log(7x+1)=\log(x-2)+1$
$\log(7x+1)-\log(x-2)=1$
We apply the quotient property:
$\log_{10}\frac{7x+1}{x-2}=1$
Next, we apply the definition of a logarithm:
$10^1=\frac{7x+1}{x-2}$
$10(x-2)=7x+1$
$10x-20=7x+1$
$10x-7x=1+20$
$3x=21$
$x=21/3$
$x=7$
We confirm that the answer works:
$\log(7\cdot7+1)=\log(7-2)+1$
$\log(50)=\log(5)+1$
$1.7=0.7+1$
$1.7=1.7$
