Solving Exponential Equations With Logarithms A Step By Step Guide

Are you struggling with exponential equations? Do you find yourself scratching your head when trying to isolate the variable nestled within the exponent? Fear not, my friends! This guide will walk you through the process of solving exponential equations using logarithms, making even the most daunting problems seem manageable. We'll break down the steps, explain the logic, and work through an example problem together. So, buckle up and let's dive into the world of logarithms!

Understanding Exponential Equations and Logarithms

Before we tackle the equation $3 e^{7 x+7}-9=30$, let's first understand the basics. Exponential equations are equations where the variable appears in the exponent. They often take the form $a^x = b$, where 'a' is the base, 'x' is the exponent, and 'b' is the result. To solve for 'x', we need a way to "undo" the exponentiation. This is where logarithms come into play.

A logarithm is the inverse operation of exponentiation. Think of it as asking the question: "To what power must I raise the base 'a' to get the result 'b'?" Mathematically, we write this as $\log_a b = x$, which is equivalent to $a^x = b$. There are two common types of logarithms:

• Common logarithm: This has a base of 10, written as $\log_{10} b$ or simply $\log b$.
• Natural logarithm: This has a base of 'e' (Euler's number, approximately 2.71828), written as $\\ln b$.

The natural logarithm is particularly useful when dealing with exponential functions involving 'e', like the one in our example. The key relationship we'll use is that $\\ln(e^x) = x$. This allows us to bring the exponent down and solve for the variable.

Key steps to solving exponential equations using logarithms:

1. Isolate the exponential term: Get the term with the exponent by itself on one side of the equation. This might involve adding, subtracting, multiplying, or dividing both sides of the equation.
2. Take the logarithm of both sides: Apply the appropriate logarithm (usually the natural logarithm if the base is 'e') to both sides of the equation. This is a crucial step that allows us to use the inverse relationship between logarithms and exponentiation.
3. Apply the power rule of logarithms: The power rule states that $\log_a (b^c) = c \log_a b$. This allows us to bring the exponent down as a coefficient.
4. Solve for the variable: After applying the power rule, you should have a linear equation that you can easily solve using basic algebraic techniques.
5. Check your solution: It's always a good idea to plug your solution back into the original equation to make sure it works and that you haven't introduced any errors.

Solving the Equation: $3 e^{7 x+7}-9=30$

Okay, guys, let's get our hands dirty and solve this equation step-by-step. We'll follow the five key steps we just outlined.

Step 1: Isolate the exponential term

Our goal is to get the term $3e^{7x+7}$ by itself. To do this, we'll first add 9 to both sides of the equation:

$$3 e^{7 x+7}-9 + 9 = 30 + 9$$

$$3 e^{7 x+7} = 39$$

Next, we'll divide both sides by 3:

$$\frac{3 e^{7 x+7}}{3} = \frac{39}{3}$$

$$e^{7 x+7} = 13$$

Great! We've successfully isolated the exponential term.

Step 2: Take the logarithm of both sides

Since the base of our exponential term is 'e', we'll use the natural logarithm (\ln). We apply the natural logarithm to both sides of the equation:

$$\\ln(e^{7 x+7}) = \\ln(13)$$

Step 3: Apply the power rule of logarithms

This is where the magic happens! The power rule allows us to bring the exponent down as a coefficient:

$$(7x + 7) \\ln(e) = \\ln(13)$$

Remember that $\\ln(e) = 1$, so we can simplify this to:

$$7x + 7 = \\ln(13)$$

Step 4: Solve for the variable

Now we have a simple linear equation. Let's solve for 'x'. First, subtract 7 from both sides:

$$7x + 7 - 7 = \\ln(13) - 7$$

$$7x = \\ln(13) - 7$$

Finally, divide both sides by 7:

$$x = \frac{\\ln(13) - 7}{7}$$

This is our solution! We can leave it in this exact form, or we can use a calculator to approximate the value:

$$x \approx \frac{2.5649 - 7}{7} \approx \frac{-4.4351}{7} \approx -0.6336$$

Step 5: Check your solution

To make sure we haven't made any mistakes, let's plug our solution back into the original equation:

$$3 e^{7(-0.6336)+7}-9=30$$

$$3 e^{-4.4352+7}-9=30$$

$$3 e^{2.5648}-9 \approx 3(13)-9 \approx 39-9 \approx 30$$

The left side of the equation is approximately equal to the right side, so our solution is correct!

Common Mistakes and How to Avoid Them

Solving exponential equations can be tricky, so let's discuss some common mistakes and how to avoid them:

• Forgetting to isolate the exponential term: This is a crucial first step. If you don't isolate the exponential term, you can't apply the logarithm effectively.
• Applying the logarithm incorrectly: Make sure you take the logarithm of both sides of the equation. Don't just take the logarithm of one term.
• Misusing the power rule: The power rule only applies when the entire term inside the logarithm is raised to a power. Be careful not to apply it incorrectly.
• Making arithmetic errors: Simple arithmetic errors can throw off your entire solution. Double-check your calculations at each step.
• Not checking your solution: Always check your solution by plugging it back into the original equation. This will help you catch any mistakes you might have made.

Practice Problems

Now that you've learned the process, let's try a few practice problems:

1. $$5^{2x} = 25$$

2. $$2 e^{x-1} = 10$$

3. $$4^{x+2} = 64$$

Work through these problems using the steps we've outlined. Remember to isolate the exponential term, take the logarithm of both sides, apply the power rule, solve for the variable, and check your solution.

Conclusion

Solving exponential equations using logarithms might seem daunting at first, but with practice and a clear understanding of the steps involved, you can master this skill. Remember to isolate the exponential term, take the logarithm of both sides, apply the power rule, solve for the variable, and check your solution. By following these steps and avoiding common mistakes, you'll be solving exponential equations like a pro in no time! Keep practicing, and you'll get the hang of it. You got this, guys!

Now, go forth and conquer those exponential equations!