Solve the exponential equation. e 2 x + 5 = 8 e^{2x+5}=8 e 2 x + 5 = 8

In this problem, we're asked to solve the exponential equation, and the exponential equation we're given is e to the power of 2x+5 is equal to 8. So let's go ahead and start with step 1 and isolate our exponential expression. Now our exponential expression here is actually already isolated, so we can go ahead and move on to step 2 and determine what log we're going to be taking. Now remember, we're going to check if it's base 10 or not base 10, and here we have a base of e, so we're going to be using the natural log. So going ahead and taking the natural log of both sides, I have the natural log of e to the power of 2x+5, and that's equal to the natural log of 8. Now that we've completed step 2, we can move on to step 3 and use our log rules to get x out of our exponent. Now here, we're gonna go ahead and pull all of this, this 2x+5, to the front here. So this becomes 2x+5 times the natural log of e, and that's equal to the natural log of 8. Now there's one other log rule we can apply here, and that's since we have this natural log, we know that natural log is just log base e. And since we're taking the natural log of e, this will just cancel out to be 1. So we're simply left with 2x+5, and that's equal to the natural log of 8. Now, moving on to step 4, we wanna go ahead and completely solve for x to get our answer. Now, in order to do that, we wanna isolate x. So we're gonna go ahead and subtract 5 from both sides first. So subtracting 5, that will cancel, leaving me with 2x is equal to the natural log of 8. I'm I'm gonna go ahead and put parentheses around that, minus 5. Now remember to be careful here. We're not subtracting this 5 from the 8 because we're taking the natural log of that 8 and then subtracting 5 from that. So just be careful with how you're writing this and how you eventually put it into your calculator. So we have one last thing to do here in isolating x, and that is dividing both sides by 2. So we're left with x is equal to the natural log of 8 minuteus 5 divided by 2. Now, we can go ahead and approximate this using our calculator for step 5 and plugging this into your calculator, making sure you're using parentheses where appropriate. We're going to end up rounding to the nearest 100ths place, getting negative 1.46 as our final answer. So x is equal to negative 1.46. Thanks for watching, and let me know if you have questions.

0. Functions

Exponential & Logarithmic Equations

Calculus

0. Functions

Exponential & Logarithmic Equations

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Multiple Choice

Solve the exponential equation.
$e^{2x+5}=8$

$x = - 1.46$

$x = - 1.11$

$x = - 0.22$

$x equals 1.39$

Verified step by step guidance

Start by isolating the exponential term. The given equation is e^{2x+5} = 8. To isolate the exponential term, divide both sides by e^5, resulting in e^{2x} = 8/e^5.

Next, take the natural logarithm of both sides to solve for the exponent. Apply the natural logarithm (ln) to both sides: ln(e^{2x}) = ln(8/e^5).

Utilize the property of logarithms that ln(e^a) = a. This simplifies the left side to 2x, so the equation becomes 2x = ln(8) - 5.

Solve for x by dividing both sides of the equation by 2. This gives x = (ln(8) - 5) / 2.

Evaluate the expression using a calculator to find the approximate value of x. Remember, the final result should be one of the given options: x = -1.46, x = -1.11, x = -0.22, or x = 1.39.