Solving Ln(4x + 4) = 5 Step By Step Guide

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Hey there, math enthusiasts! Today, we're diving deep into solving the equation ln(4x + 4) = 5. This type of problem often pops up in algebra and calculus, and mastering it is crucial for your mathematical journey. We'll break it down step by step, ensuring you not only get the correct answer but also understand the underlying principles involved. Let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what we're dealing with. The equation ln(4x + 4) = 5 involves a natural logarithm (ln). Remember, the natural logarithm is the logarithm to the base e, where e is Euler's number, approximately 2.71828. So, the equation is essentially asking: "To what power must we raise e to get 4x + 4, if that power equals 5?"

Our goal is to isolate x. To do this, we'll need to "undo" the natural logarithm. The inverse operation of the natural logarithm is exponentiation with base e. This means that if we have ln(a) = b, then e^b = a. This inverse relationship is the key to solving our equation.

The equation ln(4x + 4) = 5 might seem intimidating at first glance, but it's really just a puzzle waiting to be solved. By understanding the properties of logarithms and exponents, we can systematically unravel it. We'll first tackle the logarithm by using its inverse, and then we'll employ basic algebraic manipulations to isolate x. Think of it as peeling an onion – layer by layer, we'll get to the core of the solution.

Step-by-Step Solution

Now, let's walk through the solution step-by-step:

1. Exponentiate Both Sides

The first step is to get rid of the natural logarithm. To do this, we'll exponentiate both sides of the equation with the base e. This means we raise e to the power of each side:

e ^ ln(4x + 4) = e ^ 5

Remember the inverse relationship between the natural logarithm and exponentiation? The e and ln cancel each other out on the left side, leaving us with:

4x + 4 = e ^ 5

This is a significant simplification. We've successfully eliminated the logarithm and now have a more manageable algebraic equation.

2. Isolate the Term with x

Our next goal is to isolate the term containing x (which is 4x). To do this, we need to get rid of the +4 on the left side. We can accomplish this by subtracting 4 from both sides of the equation:

4x + 4 - 4 = e ^ 5 - 4

This simplifies to:

4x = e ^ 5 - 4

We're getting closer! The x term is now almost completely isolated.

3. Solve for x

The final step is to isolate x completely. Since x is being multiplied by 4, we'll divide both sides of the equation by 4:

4x / 4 = (e ^ 5 - 4) / 4

This gives us the exact solution for x:

x = (e ^ 5 - 4) / 4

Congratulations! We've found the exact solution. But sometimes, we need a numerical approximation. Let's move on to that.

Finding the Exact Solution

We've already arrived at the exact solution in the previous section: x = (e ^ 5 - 4) / 4. This is the most precise answer, and it's often the preferred form in many mathematical contexts. However, it's a bit abstract. To get a better sense of the value of x, we need to calculate its numerical approximation.

The exact solution, x = (e ^ 5 - 4) / 4, represents a specific point on the number line. It's the precise value that, when plugged back into the original equation, will make the equation true. However, for practical purposes, such as comparing this value with others or using it in real-world applications, a decimal approximation is often more useful.

Approximating the Solution to 4 Decimal Places

To get the solution rounded to 4 decimal places, we'll use a calculator to evaluate the expression (e ^ 5 - 4) / 4. Make sure your calculator is in the correct mode (usually radians for trigonometric functions, but this problem doesn't involve those).

  1. First, calculate e ^ 5. This should give you approximately 148.4131591.
  2. Next, subtract 4 from the result: 148.4131591 - 4 = 144.4131591.
  3. Finally, divide by 4: 144.4131591 / 4 = 36.10328978.

Now, we round this result to 4 decimal places. The fifth decimal place is 8, which is greater than or equal to 5, so we round up the fourth decimal place:

x ≈ 36.1033

Therefore, the solution to the equation ln(4x + 4) = 5, rounded to 4 decimal places, is approximately 36.1033.

This approximation gives us a concrete numerical value for x. We can now say with confidence that x is a little over 36. This kind of approximation is vital in many applications, from engineering to finance, where we need to work with practical numbers rather than abstract expressions.

Verification

It's always a good idea to verify your solution, especially in mathematics. To verify our solution, we'll plug our approximate value of x (36.1033) back into the original equation and see if it holds true:

ln(4 * 36.1033 + 4) ≈ 5

Let's calculate the left side:

  1. Multiply 4 by 36.1033: 4 * 36.1033 = 144.4132
  2. Add 4: 144.4132 + 4 = 148.4132
  3. Take the natural logarithm: ln(148.4132) ≈ 5.000002

The result is very close to 5! The slight discrepancy is due to the rounding we did earlier. If we used a more precise value of x, we'd get even closer to 5. This verification step gives us confidence that our solution is correct.

Key Takeaways

Let's recap the key steps we took to solve the equation:

  1. Exponentiated both sides to eliminate the natural logarithm.
  2. Isolate the term with x by performing algebraic manipulations.
  3. Solve for x by dividing both sides by the coefficient of x.
  4. Approximated the solution to 4 decimal places using a calculator.
  5. Verified the solution by plugging it back into the original equation.

By following these steps, you can confidently tackle similar equations involving logarithms. Remember, practice makes perfect, so try solving other problems to reinforce your understanding.

Common Mistakes to Avoid

When solving equations involving logarithms, there are a few common pitfalls to watch out for:

  • Forgetting the order of operations: Make sure you follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions.
  • Incorrectly applying the inverse relationship: Remember that the inverse of ln(x) is e^x, not 1/ln(x).
  • Rounding too early: Rounding intermediate results can lead to inaccuracies in the final answer. It's best to round only at the very end.
  • Not verifying the solution: Always check your answer by plugging it back into the original equation.

By being aware of these common mistakes, you can increase your accuracy and avoid errors.

Practice Problems

To solidify your understanding, try solving these similar equations:

  1. ln(2x + 1) = 3
  2. ln(5x - 2) = 7
  3. ln(x + 10) = 2.5

Work through these problems step-by-step, and don't hesitate to review the solution we worked through earlier. The more you practice, the more comfortable you'll become with solving logarithmic equations.

Conclusion

We've successfully solved the equation ln(4x + 4) = 5, found the exact solution, approximated it to 4 decimal places, and verified our answer. By understanding the properties of logarithms and exponents, and by following a systematic approach, you can conquer these types of problems with ease. Keep practicing, and you'll become a logarithm-solving pro in no time! Remember, math is like a muscle – the more you use it, the stronger it gets. So keep flexing those mathematical muscles!

The exact solution (using exponents) is x = (e ^ 5 - 4) / 4.

The solution, rounded to 4 decimal places is x ≈ 36.1033.