Question

1. solve each equation.

a. $3x = \log_{6} 216$
b. $\log_{9} (3x) = \log_{9} (2x + 1)$
c. $\log_{7} (5 - x) = \log_{7} (5)$
d. $\log_{5} (-3x) = 1$
e. $\log_{2} (x - 5) = \log_{2} (3)$
f. $\log_{9} (x - 1) = \log_{9} (2x)$

Explanation:

Step1: Solve for $\log_6 216$

Since $6^3=216$, $\log_6 216=3$

Step2: Solve for $x$ (part a)

$3x=3 \implies x=\frac{3}{3}=1$

Step3: Set arguments equal (part b)

For $\log_b A=\log_b B$, $A=B$: $3x=2x+1$

Step4: Solve for $x$ (part b)

$3x-2x=1 \implies x=1$

Step5: Set arguments equal (part c)

$5-x=5$

Step6: Solve for $x$ (part c)

$-x=5-5 \implies x=0$

Step7: Rewrite log to exponential (part d)

$\log_5 (-3x)=1 \implies 5^1=-3x$

Step8: Solve for $x$ (part d)

$5=-3x \implies x=-\frac{5}{3}$

Step9: Set arguments equal (part e)

$x-5=3$

Step10: Solve for $x$ (part e)

$x=3+5=8$

Step11: Set arguments equal (part f)

$x-1=2x$

Step12: Solve for $x$ (part f)

$-1=2x-x \implies x=-1$ (Note: Check domain: $x-1=-2<0$, $2x=-2<0$, so this solution is invalid; no solution exists)

Answer:

a. $x=1$
b. $x=1$
c. $x=0$
d. $x=-\frac{5}{3}$
e. $x=8$
f. No solution