Question

2a.6 exponential equations iw
equivalence property if $a^m = a^n$, then $m = n$
same base
$6^{-2m} = 6^2$ $3^{3 - 2m} = 3^4$ $5^{-x} = 5^3$ $3^{-x - 1} = 3^{-x}$ $4^{-3b - 3} = 4^3$
one change of base
$5^{3 - 3r} = 25$ $6^{-2m} = 36$ $243 = 3^{-9x}$ $125 = 5^{2x - 3}$ $16 = 4^{3x + 2}$

Explanation:

Same Base Problems

Step1: Set exponents equal

For $6^{-2m}=6^2$, apply equivalence property:
$-2m = 2$

Step2: Solve for m

$m = \frac{2}{-2} = -1$

Step1: Set exponents equal

For $3^{3-2m}=3^4$, apply equivalence property:
$3-2m = 4$

Step2: Isolate term with m

$-2m = 4-3 = 1$

Step3: Solve for m

$m = \frac{1}{-2} = -\frac{1}{2}$

Step1: Set exponents equal

For $5^{-x}=5^3$, apply equivalence property:
$-x = 3$

Step2: Solve for x

$x = -3$

Step1: Set exponents equal

For $3^{-x-1}=3^{-x}$, apply equivalence property:
$-x-1 = -x$

Step2: Simplify equation

$-x + x = 1 \implies 0=1$
This is a contradiction, so no solution exists.

Step1: Set exponents equal

For $4^{-3b-3}=4^3$, apply equivalence property:
$-3b-3 = 3$

Step2: Isolate term with b

$-3b = 3+3 = 6$

Step3: Solve for b

$b = \frac{6}{-3} = -2$

One Change of Base Problems

Step1: Rewrite 25 as $5^2$

For $5^{3-3r}=25$, substitute:
$5^{3-3r}=5^2$

Step2: Set exponents equal

$3-3r = 2$

Step3: Isolate term with r

$-3r = 2-3 = -1$

Step4: Solve for r

$r = \frac{-1}{-3} = \frac{1}{3}$

Step1: Rewrite 36 as $6^2$

For $6^{-2m}=36$, substitute:
$6^{-2m}=6^2$

Step2: Set exponents equal

$-2m = 2$

Step3: Solve for m

$m = \frac{2}{-2} = -1$

Step1: Rewrite 243 as $3^5$

For $243=3^{-9x}$, substitute:
$3^5=3^{-9x}$

Step2: Set exponents equal

$5 = -9x$

Step3: Solve for x

$x = \frac{5}{-9} = -\frac{5}{9}$

Step1: Rewrite 125 as $5^3$

For $125=5^{2x-3}$, substitute:
$5^3=5^{2x-3}$

Step2: Set exponents equal

$3 = 2x-3$

Step3: Isolate term with x

$2x = 3+3 = 6$

Step4: Solve for x

$x = \frac{6}{2} = 3$

Step1: Rewrite 16 as $4^2$

For $16=4^{3x+2}$, substitute:
$4^2=4^{3x+2}$

Step2: Set exponents equal

$2 = 3x+2$

Step3: Isolate term with x

$3x = 2-2 = 0$

Step4: Solve for x

$x = 0$

Answer:

Same Base:

1. $m=-1$
2. $m=-\frac{1}{2}$
3. $x=-3$
4. No solution
5. $b=-2$

One Change of Base:

1. $r=\frac{1}{3}$
2. $m=-1$
3. $x=-\frac{5}{9}$
4. $x=3$
5. $x=0$