Question

log (10 x+5) -log(x-4)=2​

Answer 1

Answer:

$\log _{10}\left(10x+5\right)-\log _{10}\left(x-4\right)=2\quad :\quad x=\dfrac{9}{2}\quad \left(\mathrm{Decimal}:\quad x=4.5\right)$

Step-by-step explanation:

$\log _{10}\left(10x+5\right)-\log _{10}\left(x-4\right)=2$

$\gray{\mathrm{Add\:}\log _{10}\left(x-4\right)\mathrm{\:to\:both\:sides}}$

$\log _{10}\left(10x+5\right)-\log _{10}\left(x-4\right)+\log _{10}\left(x-4\right)=2+\log _{10}\left(x-4\right)$

$\gray{\mathrm{Simplify}}$

$\log _{10}\left(10x+5\right)=2+\log _{10}\left(x-4\right)$

$\gray{\mathrm{Apply\:log\:rule}:\quad \:a=\log _b\left(b^a\right)}$

$\gray{2=\log _{10}\left(10^2\right)=\log _{10}\left(100\right)}$

$\log _{10}\left(10x+5\right)=\log _{10}\left(100\right)+\log _{10}\left(x-4\right)$

$\gray{\mathrm{Apply\:log\:rule}:\quad \log _c\left(a\right)+\log _c\left(b\right)=\log _c\left(ab\right)}$

$\gray{\log _{10}\left(100\right)+\log _{10}\left(x-4\right)=\log _{10}\left(100\left(x-4\right)\right)}$

$\log _{10}\left(10x+5\right)=\log _{10}\left(100\left(x-4\right)\right)$

$\gray{\mathrm{Apply\:log\:rule:\:\:If}\:\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\:\mathrm{then}\:f\left(x\right)=g\left(x\right)}$

$10x+5=100\left(x-4\right)$

$\gray{\mathrm{Solve\:}\:10x+5=100\left(x-4\right):\quad x=\dfrac{9}{2}}$

$x=\dfrac{9}{2}$

$\gray{\mathrm{The\:solution\:is}}$

$x=\dfrac{9}{2}$