Use natural logarithms to solve the equation.Round to the nearest thousandth.
5e^2x+11=30
Can someone explain it to me?
So far, I got.
e^2x+11=6
2x+11=In 6
So far so good. Just continue on
2x+11 = ln6
2x = ln6-11
x = (ln6 - 11)/2
ln6 is just a number, like √2 or π. Don't be afraid of using it.
Well, well, well, looks like we have an equation in disguise! Don't worry, I'm here to clown around and make it all make sense.
Let's go step by step, shall we? We want to solve the equation 5e^(2x) + 11 = 30 using natural logarithms (ln).
First, let's isolate that pesky exponential term. Subtract 11 from both sides, and we have:
5e^(2x) = 19
Now comes the fun part! We want to get rid of that exponential, so we can see the elusive x. Take the natural logarithm (ln) of both sides:
ln(5e^(2x)) = ln(19)
Now, remember that the natural logarithm of a product equals the sum of the natural logarithms of the individual terms. So we can rewrite the left side as:
ln(5) + ln(e^(2x)) = ln(19)
Now, my dear friend, you might be wondering, "What's ln(e^(2x))?" Ah, that's a easy one! The natural logarithm and the exponential function are like two peas in a pod. When they get together, they cancel each other out. So, ln(e^(2x)) simplifies to just 2x.
Now our equation is looking like this:
ln(5) + 2x = ln(19)
To solve for x, we subtract ln(5) from both sides:
2x = ln(19) - ln(5)
Finally, we can divide both sides by 2 to get:
x = (ln(19) - ln(5))/2
Now, my fellow clown enthusiast, you can plug in those logarithms, do the math, and round your answer to the nearest thousandth. Voila! You've solved the equation with the power of humor and logarithms!
To solve the equation 5e^2x + 11 = 30 using natural logarithms, you can follow these steps:
1. Begin with the equation: 5e^2x + 11 = 30.
2. Subtract 11 from both sides to isolate the exponential term: 5e^2x = 19.
3. Divide both sides by 5 to get the exponential term by itself: e^2x = 19/5.
4. Take the natural logarithm (ln) of both sides to eliminate the exponential term: ln(e^2x) = ln(19/5).
5. Use the property of logarithms which states that ln(e^y) = y: 2x = ln(19/5).
6. Divide both sides by 2 to solve for x: x = ln(19/5)/2.
To simplify the solution to the nearest thousandth, you can use a calculator or a math software. Evaluating ln(19/5)/2, we find x is approximately 0.628. Hence, the solution to the equation is x ≈ 0.628 (rounded to the nearest thousandth).
To solve the equation: 5e^(2x) + 11 = 30, you can use natural logarithms (ln) to isolate the variable x.
Here's how you can proceed:
Step 1: Subtract 11 from both sides of the equation to isolate the exponential term:
5e^(2x) = 30 - 11
5e^(2x) = 19
Step 2: Divide both sides of the equation by 5 to isolate the exponential term:
e^(2x) = 19/5
Step 3: Take the natural logarithm (ln) of both sides to remove the exponential term:
ln(e^(2x)) = ln(19/5)
2x ln(e) = ln(19/5)
Step 4: Remember that ln(e) = 1, so the equation simplifies to:
2x = ln(19/5)
Step 5: Divide both sides of the equation by 2 to solve for x:
x = ln(19/5) / 2
To find the approximate value, you can evaluate the right-hand side using a calculator:
x ≈ ln(19/5) / 2 ≈ -0.328
Therefore, the solution to the equation rounded to the nearest thousandth is x ≈ -0.328.