QuestionSolve the equation $e^{x+7}=5$ and express the solution using natural logarithms.

Studdy Solution

STEP 1

Assumptions1. The base of the natural logarithm is e (uler's number, approximately.71828) . The equation is in the form $e^{x+7}=5$
3. We need to solve for x

STEP 2

To solve for x in the given equation, we need to isolate x. We can do this by taking the natural logarithm on both sides of the equation. The natural logarithm of a number is the exponent to which the base e must be raised to produce that number. $\ln(e^{x+7})=\ln(5)$

STEP 3

The property of logarithms states that $\ln(a^b)=b\ln(a)$ . Therefore, we can simplify the left side of the equation.
$\ln(e^{x+7})=(x+7)\ln(e)$

STEP 4

Since the natural logarithm of e is1 ( $\ln(e)=1$ ), the left side of the equation simplifies to $(x+7)\ln(e) = x+7$

STEP 5

Now, we have the equation $x+7=\ln(5)$

STEP 6

To solve for x, we need to subtract from both sides of the equation.
$x=\ln(5)-$

STEP 7

This is the solution for x in terms of natural logarithms.
The solution to the equation $e^{x+7}=5$ is $x=\ln(5)-7$

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QuestionSolve the equation $e^{x+7}=5$ and express the solution using natural logarithms.

Studdy Solution

STEP 1

Assumptions1. The base of the natural logarithm is e (uler's number, approximately.71828) . The equation is in the form $e^{x+7}=5$
3. We need to solve for x

STEP 2

To solve for x in the given equation, we need to isolate x. We can do this by taking the natural logarithm on both sides of the equation. The natural logarithm of a number is the exponent to which the base e must be raised to produce that number. $\ln(e^{x+7})=\ln(5)$

STEP 3

The property of logarithms states that $\ln(a^b)=b\ln(a)$ . Therefore, we can simplify the left side of the equation.
$\ln(e^{x+7})=(x+7)\ln(e)$

STEP 4

Since the natural logarithm of e is1 ( $\ln(e)=1$ ), the left side of the equation simplifies to $(x+7)\ln(e) = x+7$

STEP 5

Now, we have the equation $x+7=\ln(5)$

STEP 6

To solve for x, we need to subtract from both sides of the equation.
$x=\ln(5)-$

STEP 7

This is the solution for x in terms of natural logarithms.
The solution to the equation $e^{x+7}=5$ is $x=\ln(5)-7$

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QuestionSolve the equation ex+7=5e^{x+7}=5ex+7=5 and express the solution using natural logarithms.

Studdy Solution

STEP 1

Assumptions1. The base of the natural logarithm is e (uler's number, approximately.71828) . The equation is in the form ex+7=5e^{x+7}=5ex+7=53. We need to solve for x

STEP 2

To solve for x in the given equation, we need to isolate x. We can do this by taking the natural logarithm on both sides of the equation. The natural logarithm of a number is the exponent to which the base e must be raised to produce that number.ln⁡(ex+7)=ln⁡(5)\ln(e^{x+7})=\ln(5)ln(ex+7)=ln(5)

STEP 3

The property of logarithms states that ln⁡(ab)=bln⁡(a)\ln(a^b)=b\ln(a)ln(ab)=bln(a). Therefore, we can simplify the left side of the equation.ln⁡(ex+7)=(x+7)ln⁡(e)\ln(e^{x+7})=(x+7)\ln(e)ln(ex+7)=(x+7)ln(e)

STEP 4

Since the natural logarithm of e is1 (ln⁡(e)=1\ln(e)=1ln(e)=1), the left side of the equation simplifies to(x+7)ln⁡(e)=x+7(x+7)\ln(e) = x+7(x+7)ln(e)=x+7

STEP 5

Now, we have the equationx+7=ln⁡(5)x+7=\ln(5)x+7=ln(5)

STEP 6

To solve for x, we need to subtract from both sides of the equation.x=ln⁡(5)−x=\ln(5)-x=ln(5)−

STEP 7

This is the solution for x in terms of natural logarithms.The solution to the equation ex+7=5e^{x+7}=5ex+7=5 isx=ln⁡(5)−7x=\ln(5)-7x=ln(5)−7

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QuestionSolve the equation $e^{x+7}=5$ and express the solution using natural logarithms.

Assumptions1. The base of the natural logarithm is e (uler's number, approximately.71828) . The equation is in the form $e^{x+7}=5$
3. We need to solve for x

To solve for x in the given equation, we need to isolate x. We can do this by taking the natural logarithm on both sides of the equation. The natural logarithm of a number is the exponent to which the base e must be raised to produce that number. $\ln(e^{x+7})=\ln(5)$

The property of logarithms states that $\ln(a^b)=b\ln(a)$ . Therefore, we can simplify the left side of the equation.
$\ln(e^{x+7})=(x+7)\ln(e)$

Since the natural logarithm of e is1 ( $\ln(e)=1$ ), the left side of the equation simplifies to $(x+7)\ln(e) = x+7$

Now, we have the equation $x+7=\ln(5)$

To solve for x, we need to subtract from both sides of the equation.
$x=\ln(5)-$

This is the solution for x in terms of natural logarithms.
The solution to the equation $e^{x+7}=5$ is $x=\ln(5)-7$