Question

সরল কর: $\log_{e}\frac{a^3b^3}{c^3}+\log_{e}\frac{b^3c^3}{d^3}+\log_{e}\frac{c^3d^3}{a^3}-3\log_{e}b^2c$

Answer 1

$\log_{e}\frac{a^3b^3}{c^3}+\log_{e}\frac{b^3c^3}{d^3}+\log_{e}\frac{c^3d^3}{a^3}-3\log_{e}b^2c$

$=\log_{e}\left(\frac{a^3b^3}{c^3}\times\frac{b^3c^3}{d^3}\times\frac{c^3d^3}{a^3}\right)-3\log_{e}b^2c$

$=\log_{e}b^6c^3-3\log_{e}b^2c$

$=\log_{e}\left(b^2c\right)^3-3\log_{e}b^2c$

$=3\log_{e}b^2c-3\log_{e}b^2c$

$=0$ [Answer]

Comments

Rules Applied:

• $\log_{a}M+log_{a}N=\log_{a}M \times N$

Answer 2

$\log_{e}\frac{a^3b^3}{c^3}+\log_{e}\frac{b^3c^3}{d^3}+\log_{e}\frac{c^3d^3}{a^3}-3\log_{e}b^2c$

$=\log_{e}\frac{a^3b^3}{c^3}+\log_{e}\frac{b^3c^3}{d^3}+\log_{e}\frac{c^3d^3}{a^3}-\log_{e}\left(b^2c\right)^3$

$=\log_{e}\frac{a^3b^3}{c^3}+\log_{e}\frac{b^3c^3}{d^3}+\log_{e}\frac{c^3d^3}{a^3}-\log_{e}b^6c^3$

$=\log_{e}\left(\frac{a^3b^3}{c^3}\times\frac{b^3c^3}{d^3}\times\frac{c^3d^3}{a^3}\div b^6c^3\right)$

$=\log_{e}\left(\frac{a^3b^3}{c^3}\times\frac{b^3c^3}{d^3}\times\frac{c^3d^3}{a^3}\times\frac{1}{b^6c^3}\right)$

$=\log_{e}1$

$=0$ [Answer]

Comments

Rules Applied:

• $\log_{a}M+\log_{a}N=\log_{a}M \times N=\log_{a}MN$
• $\log_{a}M-\log_{a}N=\log_{a}\left(M \div N\right)=\log_{a}\frac{M}{N}$
• $\log_{a}1=0$