Question

solve the equation.\\(\log_{17}(2 - 4m) = \log_{17} (-5m - 4)\\)\
a \\(\\{-6\\}\\)\
b \\(\\{5\\}\\)\
c no solution.\
d \\(\left\\{ -\dfrac{7}{18} \
ight\\}\\)

Explanation:

Step1: Use the property of logarithms

If $\log_a b = \log_a c$, then $b = c$ (for $a>0,a
eq1,b>0,c>0$). So we set $2 - 4m=-5m - 4$.

Step2: Solve the linear equation

Add $5m$ to both sides: $2 - 4m+5m=-5m - 4+5m$, which simplifies to $2 + m=-4$.
Subtract 2 from both sides: $m=-4 - 2=-6$.

Step3: Check the domain of the logarithm

For $\log_{17}(2 - 4m)$, when $m = - 6$, $2-4\times(-6)=2 + 24 = 26>0$.
For $\log_{17}(-5m - 4)$, when $m=-6$, $-5\times(-6)-4 = 30 - 4=26>0$. Both arguments are positive, so $m = - 6$ is valid.

Answer:

A. $\{-6\}$