Solved on Jan 10, 2024

Solve the exponential equations. Verify solutions using graphing technology. a) $4^{3x} = 8^{x-3}$ b) $27^{x} = 9^{2x-3}$ c) $125^{2y-1} = 25^{y+4}$ d) $16^{2k-3} = 32^{k+3}$

STEP 1

Assumptions
1. We are solving exponential equations of the form $a^{f(x)} = b^{g(x)}$ .
2. We will use the property that if $a^{f(x)} = a^{g(x)}$ , then $f(x) = g(x)$ .
3. We will also use the property that $a^{m} = (a^{n})^{m/n}$ .
4. We will check our answers using graphing technology as instructed.

STEP 2

Solve equation a) $4^{3x} = 8^{x-3}$ .
First, express both sides of the equation with a common base. The number 8 can be written as $2^3$ , and the number 4 can be written as $2^2$ .

STEP 3

Rewrite the equation with the common base of 2.
$4^{3x} = (2^2)^{3x} = 2^{6x}$ $8^{x-3} = (2^3)^{x-3} = 2^{3(x-3)}$

STEP 4

Now the equation is $2^{6x} = 2^{3(x-3)}$ .
Using the property that if $a^{f(x)} = a^{g(x)}$ , then $f(x) = g(x)$ , we can equate the exponents.
$6x = 3(x-3)$

STEP 5

Solve for $x$ .
$6x = 3x - 9$

STEP 6

Subtract $3x$ from both sides.
$6x - 3x = -9$
$3x = -9$

STEP 7

Divide both sides by 3.
$x = \frac{-9}{3}$
$x = -3$
The solution for equation a) is $x = -3$ .

STEP 8

Solve equation b) $27^{x} = 9^{2x-3}$ .
First, express both sides of the equation with a common base. The number 27 can be written as $3^3$ , and the number 9 can be written as $3^2$ .

STEP 9

Rewrite the equation with the common base of 3.
$27^{x} = (3^3)^{x} = 3^{3x}$ $9^{2x-3} = (3^2)^{2x-3} = 3^{4x-6}$

STEP 10

Now the equation is $3^{3x} = 3^{4x-6}$ .
Again, equate the exponents.
$3x = 4x - 6$

STEP 11

Subtract $4x$ from both sides.
$3x - 4x = -6$
$-x = -6$

STEP 12

Divide both sides by -1.
$x = \frac{-6}{-1}$
$x = 6$
The solution for equation b) is $x = 6$ .

STEP 13

Solve equation c) $125^{2y-1} = 25^{y+4}$ .
First, express both sides of the equation with a common base. The number 125 can be written as $5^3$ , and the number 25 can be written as $5^2$ .

STEP 14

Rewrite the equation with the common base of 5.
$125^{2y-1} = (5^3)^{2y-1} = 5^{6y-3}$ $25^{y+4} = (5^2)^{y+4} = 5^{2y+8}$

STEP 15

Now the equation is $5^{6y-3} = 5^{2y+8}$ .
Equate the exponents.
$6y - 3 = 2y + 8$

STEP 16

Subtract $2y$ from both sides.
$6y - 2y = 8 + 3$
$4y = 11$

STEP 17

Divide both sides by 4.
$y = \frac{11}{4}$
The solution for equation c) is $y = \frac{11}{4}$ .

STEP 18

Solve equation d) $16^{2k-3} = 32^{k+3}$ .
First, express both sides of the equation with a common base. The number 16 can be written as $2^4$ , and the number 32 can be written as $2^5$ .

STEP 19

Rewrite the equation with the common base of 2.
$16^{2k-3} = (2^4)^{2k-3} = 2^{8k-12}$ $32^{k+3} = (2^5)^{k+3} = 2^{5k+15}$

STEP 20

Now the equation is $2^{8k-12} = 2^{5k+15}$ .
Equate the exponents.
$8k - 12 = 5k + 15$

STEP 21

Subtract $5k$ from both sides.
$8k - 5k = 15 + 12$
$3k = 27$

STEP 22

Divide both sides by 3.
$k = \frac{27}{3}$
$k = 9$
The solution for equation d) is $k = 9$ .

STEP 23

Use graphing technology to check the solutions.
For each equation, graph both sides as separate functions and find the points of intersection. The x-coordinates of the points of intersection should correspond to the solutions found algebraically.

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Solve the exponential equations. Verify solutions using graphing technology. a) 43x=8x−34^{3x} = 8^{x-3}43x=8x−3 b) 27x=92x−327^{x} = 9^{2x-3}27x=92x−3 c) 1252y−1=25y+4125^{2y-1} = 25^{y+4}1252y−1=25y+4 d) 162k−3=32k+316^{2k-3} = 32^{k+3}162k−3=32k+3

STEP 1

STEP 2

Solve equation a) 43x=8x−34^{3x} = 8^{x-3}43x=8x−3.First, express both sides of the equation with a common base. The number 8 can be written as 232^323, and the number 4 can be written as 222^222.

STEP 3

Rewrite the equation with the common base of 2.43x=(22)3x=26x4^{3x} = (2^2)^{3x} = 2^{6x}43x=(22)3x=26x 8x−3=(23)x−3=23(x−3)8^{x-3} = (2^3)^{x-3} = 2^{3(x-3)}8x−3=(23)x−3=23(x−3)

STEP 4

Now the equation is 26x=23(x−3)2^{6x} = 2^{3(x-3)}26x=23(x−3).Using the property that if af(x)=ag(x)a^{f(x)} = a^{g(x)}af(x)=ag(x), then f(x)=g(x)f(x) = g(x)f(x)=g(x), we can equate the exponents.6x=3(x−3)6x = 3(x-3)6x=3(x−3)

STEP 5

Solve for xxx.6x=3x−96x = 3x - 96x=3x−9

STEP 6

Subtract 3x3x3x from both sides.6x−3x=−96x - 3x = -96x−3x=−93x=−93x = -93x=−9

STEP 7

Divide both sides by 3.x=−93x = \frac{-9}{3}x=3−9​x=−3x = -3x=−3The solution for equation a) is x=−3x = -3x=−3.

STEP 8

Solve equation b) 27x=92x−327^{x} = 9^{2x-3}27x=92x−3.First, express both sides of the equation with a common base. The number 27 can be written as 333^333, and the number 9 can be written as 323^232.

STEP 9

Rewrite the equation with the common base of 3.27x=(33)x=33x27^{x} = (3^3)^{x} = 3^{3x}27x=(33)x=33x 92x−3=(32)2x−3=34x−69^{2x-3} = (3^2)^{2x-3} = 3^{4x-6}92x−3=(32)2x−3=34x−6

STEP 10

Now the equation is 33x=34x−63^{3x} = 3^{4x-6}33x=34x−6.Again, equate the exponents.3x=4x−63x = 4x - 63x=4x−6

STEP 11

Subtract 4x4x4x from both sides.3x−4x=−63x - 4x = -63x−4x=−6−x=−6-x = -6−x=−6

STEP 12

Divide both sides by -1.x=−6−1x = \frac{-6}{-1}x=−1−6​x=6x = 6x=6The solution for equation b) is x=6x = 6x=6.

STEP 13

Solve equation c) 1252y−1=25y+4125^{2y-1} = 25^{y+4}1252y−1=25y+4.First, express both sides of the equation with a common base. The number 125 can be written as 535^353, and the number 25 can be written as 525^252.

STEP 14

Rewrite the equation with the common base of 5.1252y−1=(53)2y−1=56y−3125^{2y-1} = (5^3)^{2y-1} = 5^{6y-3}1252y−1=(53)2y−1=56y−3 25y+4=(52)y+4=52y+825^{y+4} = (5^2)^{y+4} = 5^{2y+8}25y+4=(52)y+4=52y+8

STEP 15

Now the equation is 56y−3=52y+85^{6y-3} = 5^{2y+8}56y−3=52y+8.Equate the exponents.6y−3=2y+86y - 3 = 2y + 86y−3=2y+8

STEP 16

Subtract 2y2y2y from both sides.6y−2y=8+36y - 2y = 8 + 36y−2y=8+34y=114y = 114y=11

STEP 17

Divide both sides by 4.y=114y = \frac{11}{4}y=411​The solution for equation c) is y=114y = \frac{11}{4}y=411​.

STEP 18

Solve equation d) 162k−3=32k+316^{2k-3} = 32^{k+3}162k−3=32k+3.First, express both sides of the equation with a common base. The number 16 can be written as 242^424, and the number 32 can be written as 252^525.

STEP 19

Rewrite the equation with the common base of 2.162k−3=(24)2k−3=28k−1216^{2k-3} = (2^4)^{2k-3} = 2^{8k-12}162k−3=(24)2k−3=28k−12 32k+3=(25)k+3=25k+1532^{k+3} = (2^5)^{k+3} = 2^{5k+15}32k+3=(25)k+3=25k+15

STEP 20

Now the equation is 28k−12=25k+152^{8k-12} = 2^{5k+15}28k−12=25k+15.Equate the exponents.8k−12=5k+158k - 12 = 5k + 158k−12=5k+15

STEP 21

Subtract 5k5k5k from both sides.8k−5k=15+128k - 5k = 15 + 128k−5k=15+123k=273k = 273k=27

STEP 22

Divide both sides by 3.k=273k = \frac{27}{3}k=327​k=9k = 9k=9The solution for equation d) is k=9k = 9k=9.

STEP 23

Use graphing technology to check the solutions.For each equation, graph both sides as separate functions and find the points of intersection. The x-coordinates of the points of intersection should correspond to the solutions found algebraically.

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Solve the exponential equations. Verify solutions using graphing technology. a) $4^{3x} = 8^{x-3}$ b) $27^{x} = 9^{2x-3}$ c) $125^{2y-1} = 25^{y+4}$ d) $16^{2k-3} = 32^{k+3}$

Solve equation a) $4^{3x} = 8^{x-3}$ .
First, express both sides of the equation with a common base. The number 8 can be written as $2^3$ , and the number 4 can be written as $2^2$ .

Rewrite the equation with the common base of 2.
$4^{3x} = (2^2)^{3x} = 2^{6x}$ $8^{x-3} = (2^3)^{x-3} = 2^{3(x-3)}$

Now the equation is $2^{6x} = 2^{3(x-3)}$ .
Using the property that if $a^{f(x)} = a^{g(x)}$ , then $f(x) = g(x)$ , we can equate the exponents.
$6x = 3(x-3)$

Solve for $x$ .
$6x = 3x - 9$

Subtract $3x$ from both sides.
$6x - 3x = -9$
$3x = -9$

Divide both sides by 3.
$x = \frac{-9}{3}$
$x = -3$
The solution for equation a) is $x = -3$ .

Solve equation b) $27^{x} = 9^{2x-3}$ .
First, express both sides of the equation with a common base. The number 27 can be written as $3^3$ , and the number 9 can be written as $3^2$ .

Rewrite the equation with the common base of 3.
$27^{x} = (3^3)^{x} = 3^{3x}$ $9^{2x-3} = (3^2)^{2x-3} = 3^{4x-6}$

Now the equation is $3^{3x} = 3^{4x-6}$ .
Again, equate the exponents.
$3x = 4x - 6$

Subtract $4x$ from both sides.
$3x - 4x = -6$
$-x = -6$

Divide both sides by -1.
$x = \frac{-6}{-1}$
$x = 6$
The solution for equation b) is $x = 6$ .

Solve equation c) $125^{2y-1} = 25^{y+4}$ .
First, express both sides of the equation with a common base. The number 125 can be written as $5^3$ , and the number 25 can be written as $5^2$ .

Rewrite the equation with the common base of 5.
$125^{2y-1} = (5^3)^{2y-1} = 5^{6y-3}$ $25^{y+4} = (5^2)^{y+4} = 5^{2y+8}$

Now the equation is $5^{6y-3} = 5^{2y+8}$ .
Equate the exponents.
$6y - 3 = 2y + 8$

Subtract $2y$ from both sides.
$6y - 2y = 8 + 3$
$4y = 11$

Divide both sides by 4.
$y = \frac{11}{4}$
The solution for equation c) is $y = \frac{11}{4}$ .

Solve equation d) $16^{2k-3} = 32^{k+3}$ .
First, express both sides of the equation with a common base. The number 16 can be written as $2^4$ , and the number 32 can be written as $2^5$ .

Rewrite the equation with the common base of 2.
$16^{2k-3} = (2^4)^{2k-3} = 2^{8k-12}$ $32^{k+3} = (2^5)^{k+3} = 2^{5k+15}$

Now the equation is $2^{8k-12} = 2^{5k+15}$ .
Equate the exponents.
$8k - 12 = 5k + 15$

Subtract $5k$ from both sides.
$8k - 5k = 15 + 12$
$3k = 27$

Divide both sides by 3.
$k = \frac{27}{3}$
$k = 9$
The solution for equation d) is $k = 9$ .

Use graphing technology to check the solutions.
For each equation, graph both sides as separate functions and find the points of intersection. The x-coordinates of the points of intersection should correspond to the solutions found algebraically.