Question Solve the equation $e^{x}=e^{8 x+14}$ . The solution set is a simplified fraction or integer.

Studdy Solution

STEP 1

Assumptions
1. The given equation is $e^{x}=e^{8x+14}$ .
2. We are looking for the solution set, which means all values of $x$ that satisfy the equation.
3. The base of the exponential function is $e$ (Euler's number), which is a constant approximately equal to 2.71828.

STEP 2

The equation is $e^{x}=e^{8x+14}$ . Since the bases are the same, we can equate the exponents. This is based on the property of exponents which states that if $a^b = a^c$ , then $b = c$ for $a > 0$ and $a ≠ 1$ .
$x = 8x + 14$

STEP 3

We can solve this equation for $x$ by first subtracting $8x$ from both sides to isolate $x$ on one side of the equation.
$x - 8x = 14$

STEP 4

Simplify the left side of the equation.
$-7x = 14$

STEP 5

Next, divide both sides by $-7$ to solve for $x$ .
$x = \frac{14}{-7}$

STEP 6

Calculate the value of $x$ .
$x = -2$
The solution set is $x = -2$ .

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Question Solve the equation $e^{x}=e^{8 x+14}$ . The solution set is a simplified fraction or integer.

Studdy Solution

STEP 1

Assumptions
1. The given equation is $e^{x}=e^{8x+14}$ .
2. We are looking for the solution set, which means all values of $x$ that satisfy the equation.
3. The base of the exponential function is $e$ (Euler's number), which is a constant approximately equal to 2.71828.

STEP 2

The equation is $e^{x}=e^{8x+14}$ . Since the bases are the same, we can equate the exponents. This is based on the property of exponents which states that if $a^b = a^c$ , then $b = c$ for $a > 0$ and $a ≠ 1$ .
$x = 8x + 14$

STEP 3

We can solve this equation for $x$ by first subtracting $8x$ from both sides to isolate $x$ on one side of the equation.
$x - 8x = 14$

STEP 4

Simplify the left side of the equation.
$-7x = 14$

STEP 5

Next, divide both sides by $-7$ to solve for $x$ .
$x = \frac{14}{-7}$

STEP 6

Calculate the value of $x$ .
$x = -2$
The solution set is $x = -2$ .

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Question Solve the equation ex=e8x+14e^{x}=e^{8 x+14}ex=e8x+14. The solution set is a simplified fraction or integer.

Studdy Solution

STEP 1

Assumptions1. The given equation is ex=e8x+14e^{x}=e^{8x+14}ex=e8x+14.2. We are looking for the solution set, which means all values of xxx that satisfy the equation.3. The base of the exponential function is eee (Euler's number), which is a constant approximately equal to 2.71828.

STEP 2

The equation is ex=e8x+14e^{x}=e^{8x+14}ex=e8x+14. Since the bases are the same, we can equate the exponents. This is based on the property of exponents which states that if ab=aca^b = a^cab=ac, then b=cb = cb=c for a>0a > 0a>0 and a≠1a ≠ 1a=1.x=8x+14x = 8x + 14x=8x+14

STEP 3

We can solve this equation for xxx by first subtracting 8x8x8x from both sides to isolate xxx on one side of the equation.x−8x=14x - 8x = 14x−8x=14

STEP 4

Simplify the left side of the equation.−7x=14-7x = 14−7x=14

STEP 5

Next, divide both sides by −7-7−7 to solve for xxx.x=14−7x = \frac{14}{-7}x=−714​

STEP 6

Calculate the value of xxx.x=−2x = -2x=−2The solution set is x=−2x = -2x=−2.

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Question Solve the equation $e^{x}=e^{8 x+14}$ . The solution set is a simplified fraction or integer.

Assumptions
1. The given equation is $e^{x}=e^{8x+14}$ .
2. We are looking for the solution set, which means all values of $x$ that satisfy the equation.
3. The base of the exponential function is $e$ (Euler's number), which is a constant approximately equal to 2.71828.

The equation is $e^{x}=e^{8x+14}$ . Since the bases are the same, we can equate the exponents. This is based on the property of exponents which states that if $a^b = a^c$ , then $b = c$ for $a > 0$ and $a ≠ 1$ .
$x = 8x + 14$

We can solve this equation for $x$ by first subtracting $8x$ from both sides to isolate $x$ on one side of the equation.
$x - 8x = 14$

Simplify the left side of the equation.
$-7x = 14$

Next, divide both sides by $-7$ to solve for $x$ .
$x = \frac{14}{-7}$

Calculate the value of $x$ .
$x = -2$
The solution set is $x = -2$ .