Math / Algebra

QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use $e=2.71828182845905$ .

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $e$ to a power equal to 5 to a power. Watch out! Remember the properties of logarithms and exponents!
Don't mix up the bases when applying logarithmic properties.

STEP 2

1. Take Natural Logarithms
2. Apply Logarithmic Properties
3. Isolate $x$
4. Calculate the Result

STEP 3

Let's take the natural logarithm of both sides of our equation $e^{3x+13} = 5^{\frac{3x}{7}}$ .
Remember, we can do this because applying the same function to both sides of an equation maintains the equality.
We choose the natural log because it simplifies things when we have a base of $e$ . $\ln(e^{3x+13}) = \ln(5^{\frac{3x}{7}})$

STEP 4

Now, let's use the power rule of logarithms: $\ln(a^b) = b \cdot \ln(a)$ .
This rule lets us bring those exponents down in front. $(3x+13) \cdot \ln(e) = \frac{3x}{7} \cdot \ln(5)$

STEP 5

Remember, $\ln(e)$ is just 1!
So, we can simplify our equation: $3x+13 = \frac{3x}{7} \cdot \ln(5)$

STEP 6

Let's move all the terms with $x$ to one side of the equation and the constant terms to the other.
We'll subtract $3x$ from both sides: $13 = \frac{3x}{7} \cdot \ln(5) - 3x$

STEP 7

Now, we can factor out $x$ from the right side: $13 = x \cdot \left( \frac{3}{7} \cdot \ln(5) - 3 \right)$

STEP 8

To completely isolate $x$ , we'll divide both sides by $\left( \frac{3}{7} \cdot \ln(5) - 3 \right)$ : $x = \frac{13}{\frac{3}{7} \cdot \ln(5) - 3}$

STEP 9

Time to plug in the value for $\ln(5)$ which is approximately $1.6094$ . $x = \frac{13}{\frac{3}{7} \cdot 1.6094 - 3}$

STEP 10

Let's simplify the denominator: $x = \frac{13}{0.6897 - 3}$ $x = \frac{13}{-2.3103}$

STEP 11

Finally, we calculate the value of $x$ : $x \approx -5.63$

STEP 12

The exact solution is $x = \frac{13}{\frac{3}{7}\ln(5) - 3}$ .
The approximate solution, rounded to two decimal places, is $x \approx -5.63$ .

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QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use $e=2.71828182845905$ .

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $e$ to a power equal to 5 to a power. Watch out! Remember the properties of logarithms and exponents!
Don't mix up the bases when applying logarithmic properties.

STEP 2

1. Take Natural Logarithms
2. Apply Logarithmic Properties
3. Isolate $x$
4. Calculate the Result

STEP 3

STEP 4

Now, let's use the power rule of logarithms: $\ln(a^b) = b \cdot \ln(a)$ .
This rule lets us bring those exponents down in front. $(3x+13) \cdot \ln(e) = \frac{3x}{7} \cdot \ln(5)$

STEP 5

Remember, $\ln(e)$ is just 1!
So, we can simplify our equation: $3x+13 = \frac{3x}{7} \cdot \ln(5)$

STEP 6

Let's move all the terms with $x$ to one side of the equation and the constant terms to the other.
We'll subtract $3x$ from both sides: $13 = \frac{3x}{7} \cdot \ln(5) - 3x$

STEP 7

Now, we can factor out $x$ from the right side: $13 = x \cdot \left( \frac{3}{7} \cdot \ln(5) - 3 \right)$

STEP 8

To completely isolate $x$ , we'll divide both sides by $\left( \frac{3}{7} \cdot \ln(5) - 3 \right)$ : $x = \frac{13}{\frac{3}{7} \cdot \ln(5) - 3}$

STEP 9

Time to plug in the value for $\ln(5)$ which is approximately $1.6094$ . $x = \frac{13}{\frac{3}{7} \cdot 1.6094 - 3}$

STEP 10

Let's simplify the denominator: $x = \frac{13}{0.6897 - 3}$ $x = \frac{13}{-2.3103}$

STEP 11

Finally, we calculate the value of $x$ : $x \approx -5.63$

STEP 12

The exact solution is $x = \frac{13}{\frac{3}{7}\ln(5) - 3}$ .
The approximate solution, rounded to two decimal places, is $x \approx -5.63$ .

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QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use e=2.71828182845905e=2.71828182845905e=2.71828182845905.

Studdy Solution

STEP 1

What is this asking? We need to find the value of xxx that makes eee to a power equal to 5 to a power. Watch out! Remember the properties of logarithms and exponents!Don't mix up the bases when applying logarithmic properties.

STEP 2

1. Take Natural Logarithms2. Apply Logarithmic Properties3. Isolate xxx4. Calculate the Result

STEP 3

STEP 4

Now, let's use the **power rule of logarithms**: ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b \cdot \ln(a)ln(ab)=b⋅ln(a).This rule lets us bring those exponents down in front. (3x+13)⋅ln⁡(e)=3x7⋅ln⁡(5) (3x+13) \cdot \ln(e) = \frac{3x}{7} \cdot \ln(5) (3x+13)⋅ln(e)=73x​⋅ln(5)

STEP 5

Remember, ln⁡(e)\ln(e)ln(e) is just 1!So, we can simplify our equation: 3x+13=3x7⋅ln⁡(5) 3x+13 = \frac{3x}{7} \cdot \ln(5) 3x+13=73x​⋅ln(5)

STEP 6

Let's **move all the terms with** xxx **to one side** of the equation and the **constant terms to the other**.We'll subtract 3x3x3x from both sides: 13=3x7⋅ln⁡(5)−3x 13 = \frac{3x}{7} \cdot \ln(5) - 3x 13=73x​⋅ln(5)−3x

STEP 7

Now, we can **factor out** xxx from the right side: 13=x⋅(37⋅ln⁡(5)−3) 13 = x \cdot \left( \frac{3}{7} \cdot \ln(5) - 3 \right) 13=x⋅(73​⋅ln(5)−3)

STEP 8

To **completely isolate** xxx, we'll **divide** both sides by (37⋅ln⁡(5)−3)\left( \frac{3}{7} \cdot \ln(5) - 3 \right)(73​⋅ln(5)−3): x=1337⋅ln⁡(5)−3 x = \frac{13}{\frac{3}{7} \cdot \ln(5) - 3} x=73​⋅ln(5)−313​

STEP 9

Time to **plug in the value** for ln⁡(5)\ln(5)ln(5) which is approximately 1.60941.60941.6094. x=1337⋅1.6094−3 x = \frac{13}{\frac{3}{7} \cdot 1.6094 - 3} x=73​⋅1.6094−313​

STEP 10

Let's **simplify the denominator**: x=130.6897−3 x = \frac{13}{0.6897 - 3} x=0.6897−313​ x=13−2.3103 x = \frac{13}{-2.3103} x=−2.310313​

STEP 11

Finally, we **calculate the value of** xxx: x≈−5.63 x \approx -5.63 x≈−5.63

STEP 12

The **exact solution** is x=1337ln⁡(5)−3x = \frac{13}{\frac{3}{7}\ln(5) - 3}x=73​ln(5)−313​.The **approximate solution**, rounded to two decimal places, is x≈−5.63x \approx -5.63x≈−5.63.

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QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use $e=2.71828182845905$ .

What is this asking? We need to find the value of $x$ that makes $e$ to a power equal to 5 to a power. Watch out! Remember the properties of logarithms and exponents!
Don't mix up the bases when applying logarithmic properties.

1. Take Natural Logarithms
2. Apply Logarithmic Properties
3. Isolate $x$
4. Calculate the Result

Now, let's use the power rule of logarithms: $\ln(a^b) = b \cdot \ln(a)$ .
This rule lets us bring those exponents down in front. $(3x+13) \cdot \ln(e) = \frac{3x}{7} \cdot \ln(5)$

Remember, $\ln(e)$ is just 1!
So, we can simplify our equation: $3x+13 = \frac{3x}{7} \cdot \ln(5)$

Let's move all the terms with $x$ to one side of the equation and the constant terms to the other.
We'll subtract $3x$ from both sides: $13 = \frac{3x}{7} \cdot \ln(5) - 3x$

Now, we can factor out $x$ from the right side: $13 = x \cdot \left( \frac{3}{7} \cdot \ln(5) - 3 \right)$

To completely isolate $x$ , we'll divide both sides by $\left( \frac{3}{7} \cdot \ln(5) - 3 \right)$ : $x = \frac{13}{\frac{3}{7} \cdot \ln(5) - 3}$

Time to plug in the value for $\ln(5)$ which is approximately $1.6094$ . $x = \frac{13}{\frac{3}{7} \cdot 1.6094 - 3}$

Let's simplify the denominator: $x = \frac{13}{0.6897 - 3}$ $x = \frac{13}{-2.3103}$

Finally, we calculate the value of $x$ : $x \approx -5.63$

The exact solution is $x = \frac{13}{\frac{3}{7}\ln(5) - 3}$ .
The approximate solution, rounded to two decimal places, is $x \approx -5.63$ .