Math Help - Logs Problem

1. Logs Problem

$N_{A}(t) = 10000 + 1000t$
$N_{C}(t) = 8000 + 3 \cdot 2^t$

Could someone please explain how $N_{A}(t) = N_{C}(t)$ if and only if $t=\displaystyle{\frac{1}{\log_{10}2}}\cdot( 3 + \log_{10}(\displaystyle{\frac{2 + t}{3})})$?

2. Re: Logs Problem

Originally Posted by Fratricide
$N_{A}(t) = 10000 + 1000t$
$N_{C}(t) = 8000 + 3 \cdot 2^t$

Could someone please explain how $N_{A}(t) = N_{C}(t)$ if and only if $t=\displaystyle{\frac{1}{\log_{10}2}}\cdot( 3 + \log_{10}(\displaystyle{\frac{2 + t}{3})})$?
$10000+1000t=8000+3\cdot 2^t$

$2000+1000t=3\cdot 2^t$

$1000(2+t)=3\cdot 2^t$

$3+\log_{10}(2+t)=\log_{10}(2)t+\log_{10}(3)$

$3+\log_{10}(2+t)-\log_{10}(3)=\log_{10}(2)t$

$3+\log_{10}\left(\dfrac{2+t}{3}\right)=\log_{10}( 2)t$

$t=\dfrac{3+\log_{10}\left(\dfrac{2+t}{3}\right)}{ \log_{10}(2)}$