Math / Algebra

QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. $3 e^{3 x+4}=65$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $3e^{3x+4}$ equal to 65, and give the answer in exact form and as a decimal. Watch out! Remember those exponent rules and be careful with the order of operations when solving for $x$ !

STEP 2

1. Isolate the exponential term
2. Apply the natural logarithm
3. Solve for $x$
4. Calculate the approximate value

STEP 3

We want to get $e^{3x+4}$ by itself.
To do this, we divide both sides of the equation by 3: $\frac{3e^{3x+4}}{3} = \frac{65}{3}$ $e^{3x+4} = \frac{65}{3}$ Now we have the exponential term isolated!

STEP 4

To get $3x+4$ out of the exponent, we can apply the natural logarithm to both sides of the equation: $\ln(e^{3x+4}) = \ln\left(\frac{65}{3}\right)$ Remember, the natural logarithm and the exponential function with base $e$ are inverse operations, so they "undo" each other!

STEP 5

Using the property $\ln(e^a) = a$ , we can simplify the left side: $3x+4 = \ln\left(\frac{65}{3}\right)$

STEP 6

Let's isolate the term with $x$ by subtracting 4 from both sides: $3x + 4 - 4 = \ln\left(\frac{65}{3}\right) - 4$ $3x = \ln\left(\frac{65}{3}\right) - 4$

STEP 7

Finally, we divide both sides by 3 to solve for $x$ : $\frac{3x}{3} = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ $x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ This is our exact solution!

STEP 8

Now, we can use a calculator to find the approximate value of $x$ : $x \approx \frac{3.054 - 4}{3}$ $x \approx \frac{-0.946}{3}$ $x \approx -0.315$ So, $x$ is approximately equal to -0.32 when rounded to two decimal places.

STEP 9

The exact solution is $x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ and the approximate solution is $x \approx -0.32$ .

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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. $3 e^{3 x+4}=65$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $3e^{3x+4}$ equal to 65, and give the answer in exact form and as a decimal. Watch out! Remember those exponent rules and be careful with the order of operations when solving for $x$ !

STEP 2

1. Isolate the exponential term
2. Apply the natural logarithm
3. Solve for $x$
4. Calculate the approximate value

STEP 3

We want to get $e^{3x+4}$ by itself.
To do this, we divide both sides of the equation by 3: $\frac{3e^{3x+4}}{3} = \frac{65}{3}$ $e^{3x+4} = \frac{65}{3}$ Now we have the exponential term isolated!

STEP 4

To get $3x+4$ out of the exponent, we can apply the natural logarithm to both sides of the equation: $\ln(e^{3x+4}) = \ln\left(\frac{65}{3}\right)$ Remember, the natural logarithm and the exponential function with base $e$ are inverse operations, so they "undo" each other!

STEP 5

Using the property $\ln(e^a) = a$ , we can simplify the left side: $3x+4 = \ln\left(\frac{65}{3}\right)$

STEP 6

Let's isolate the term with $x$ by subtracting 4 from both sides: $3x + 4 - 4 = \ln\left(\frac{65}{3}\right) - 4$ $3x = \ln\left(\frac{65}{3}\right) - 4$

STEP 7

Finally, we divide both sides by 3 to solve for $x$ : $\frac{3x}{3} = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ $x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ This is our exact solution!

STEP 8

Now, we can use a calculator to find the approximate value of $x$ : $x \approx \frac{3.054 - 4}{3}$ $x \approx \frac{-0.946}{3}$ $x \approx -0.315$ So, $x$ is approximately equal to -0.32 when rounded to two decimal places.

STEP 9

The exact solution is $x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ and the approximate solution is $x \approx -0.32$ .

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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. 3e3x+4=653 e^{3 x+4}=653e3x+4=65

Studdy Solution

STEP 1

What is this asking? We need to find the value of xxx that makes 3e3x+43e^{3x+4}3e3x+4 equal to 65, and give the answer in exact form and as a decimal. Watch out! Remember those exponent rules and be careful with the order of operations when solving for xxx!

STEP 2

1. Isolate the exponential term2. Apply the natural logarithm3. Solve for xxx4. Calculate the approximate value

STEP 3

We want to get e3x+4e^{3x+4}e3x+4 by itself.To do this, we **divide both sides** of the equation by **3**: 3e3x+43=653 \frac{3e^{3x+4}}{3} = \frac{65}{3} 33e3x+4​=365​ e3x+4=653 e^{3x+4} = \frac{65}{3} e3x+4=365​Now we have the exponential term isolated!

STEP 4

STEP 5

Using the property ln⁡(ea)=a\ln(e^a) = aln(ea)=a, we can simplify the left side: 3x+4=ln⁡(653) 3x+4 = \ln\left(\frac{65}{3}\right) 3x+4=ln(365​)

STEP 6

Let's **isolate** the term with xxx by **subtracting 4** from both sides: 3x+4−4=ln⁡(653)−4 3x + 4 - 4 = \ln\left(\frac{65}{3}\right) - 4 3x+4−4=ln(365​)−4 3x=ln⁡(653)−4 3x = \ln\left(\frac{65}{3}\right) - 4 3x=ln(365​)−4

STEP 7

Finally, we **divide both sides by 3** to solve for xxx: 3x3=ln⁡(653)−43 \frac{3x}{3} = \frac{\ln\left(\frac{65}{3}\right) - 4}{3} 33x​=3ln(365​)−4​ x=ln⁡(653)−43 x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3} x=3ln(365​)−4​This is our **exact solution**!

STEP 8

STEP 9

The exact solution is x=ln⁡(653)−43x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}x=3ln(365​)−4​ and the approximate solution is x≈−0.32x \approx -0.32x≈−0.32.

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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. $3 e^{3 x+4}=65$

What is this asking? We need to find the value of $x$ that makes $3e^{3x+4}$ equal to 65, and give the answer in exact form and as a decimal. Watch out! Remember those exponent rules and be careful with the order of operations when solving for $x$ !

1. Isolate the exponential term
2. Apply the natural logarithm
3. Solve for $x$
4. Calculate the approximate value

We want to get $e^{3x+4}$ by itself.
To do this, we divide both sides of the equation by 3: $\frac{3e^{3x+4}}{3} = \frac{65}{3}$ $e^{3x+4} = \frac{65}{3}$ Now we have the exponential term isolated!

Using the property $\ln(e^a) = a$ , we can simplify the left side: $3x+4 = \ln\left(\frac{65}{3}\right)$

Let's isolate the term with $x$ by subtracting 4 from both sides: $3x + 4 - 4 = \ln\left(\frac{65}{3}\right) - 4$ $3x = \ln\left(\frac{65}{3}\right) - 4$

Finally, we divide both sides by 3 to solve for $x$ : $\frac{3x}{3} = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ $x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ This is our exact solution!

The exact solution is $x = \frac{\ln\left(\frac{65}{3}\right) - 4}{3}$ and the approximate solution is $x \approx -0.32$ .