Question

solve. simplify your answer(s). if there are multiple answers, separate them with commas. \\(\log_{9}(-r + 3) = \log_{9}(-5r - 1)\\) \\(r = \square\\)

Explanation:

Step1: Use Logarithm Property

If $\log_b(x)=\log_b(y)$, then $x = y$ (for $b>0,b
eq1,x>0,y>0$). So we set $-r + 3=-5r - 1$.

Step2: Solve for r

Add $5r$ to both sides: $-r+5r + 3=-5r+5r - 1$ which simplifies to $4r + 3=-1$.
Subtract 3 from both sides: $4r+3 - 3=-1 - 3$ which gives $4r=-4$.
Divide both sides by 4: $r=\frac{-4}{4}=-1$.
Now we need to check the domain:
For $\log_9(-r + 3)$, when $r = -1$, $-(-1)+3=1 + 3 = 4>0$.
For $\log_9(-5r - 1)$, when $r=-1$, $-5(-1)-1 = 5 - 1 = 4>0$. So the solution is valid.

Answer:

-1