Equations involving exponentials

Introduction: Cracking the Code of Exponentials

Hi there! Welcome to one of the most satisfying parts of your A-Level Maths journey. So far, you have probably learned what an exponential looks like (like \(2^x\) or \(e^x\)), but what happens when you need to find the value of \(x\) and it is "stuck" up in the air as a power?

In this chapter, we are going to learn how to bring those variables back down to earth. Solving equations involving exponentials is a bit like being a locksmith; you just need the right tool (usually logarithms) to unlock the equation and find the answer. Don't worry if this seems tricky at first—once you see the pattern, it becomes much easier!

1. The "Same Base" Shortcut

Before we use any fancy tools, we always check if there is a shortcut. If both sides of an equation can be written using the same base, we can simply "cancel out" the bases and solve for the powers.

The Rule: If \(a^x = a^y\), then \(x = y\).

Example: Solve \(2^x = 8\).
1. We know that \(8\) is the same as \(2^3\).
2. So, we can rewrite the equation as \(2^x = 2^3\).
3. Since the bases are the same, \(x\) must be \(3\)!

Quick Review: Always look for common bases like 2, 4, 8, 16 or 3, 9, 27 before doing anything else!

2. The Secret Weapon: Taking Logs of Both Sides

Most of the time, the bases won't match. For example, how do you solve \(3^x = 20\)? You can't write 20 as a power of 3 easily. This is where we use Logarithms.

Think of it this way: Logarithms are like a "ladder" that allows the exponent to climb down and stand on the ground.

Step-by-Step: Solving \(a^x = b\)

1. Take logs of both sides: Write "log" in front of both terms: \(\log(3^x) = \log(20)\).
2. Use the Power Law: Move the \(x\) to the front: \(x \log(3) = \log(20)\).
3. Rearrange for x: Divide by \(\log(3)\): \(x = \frac{\log(20)}{\log(3)}\).
4. Calculate: Use your calculator to get the final decimal answer (usually to 3 significant figures).

Did you know? You can use log (base 10) or ln (natural log). Most students prefer ln because it's shorter to write and essential when dealing with the number \(e\)!

3. Handling More Complex Exponents

The syllabus requires you to solve equations where both sides have exponents, such as \(2^x = 3^{2x-1}\). This looks scary, but the steps are exactly the same. We just need a little bit of algebra at the end.

Step-by-Step Example: \(2^x = 3^{2x-1}\)

1. Take logs of both sides:
\(\ln(2^x) = \ln(3^{2x-1})\)

2. Bring the powers down:
\(x \ln(2) = (2x-1) \ln(3)\)

3. Expand the brackets:
\(x \ln(2) = 2x \ln(3) - \ln(3)\)

4. Collect all the 'x' terms on one side:
\(\ln(3) = 2x \ln(3) - x \ln(2)\)

5. Factorise out the 'x':
\(\ln(3) = x(2 \ln(3) - \ln(2))\)

6. Solve for x:
\(x = \frac{\ln(3)}{2 \ln(3) - \ln(2)}\)

Common Mistake to Avoid: When you bring down a power like \((2x-1)\), always put it in brackets. If you don't, you might forget to multiply both parts by the log!

Key Takeaway: No matter how messy the power looks, the process is always: Log \(\rightarrow\) Drop Power \(\rightarrow\) Rearrange \(\rightarrow\) Solve.

4. Equations with \(e^x\)

When you see the special number \(e\), you should always use the natural logarithm, ln. This is because ln and \(e\) are inverses—they effectively "cancel each other out."

The Rule: \(\ln(e^x) = x\)

Example: Solve \(e^{2x} = 10\).
1. Take natural logs: \(\ln(e^{2x}) = \ln(10)\).
2. The left side simplifies: \(2x = \ln(10)\).
3. Divide by 2: \(x = \frac{\ln(10)}{2}\).
4. Type into calculator: \(x \approx 1.15\).

5. Hidden Quadratics

Sometimes, an exponential equation is actually a quadratic equation in disguise. These often look like this: \(e^{2x} - 5e^x + 6 = 0\).

The Trick: Substitution
If we let \(u = e^x\), then \(u^2 = (e^x)^2 = e^{2x}\).

Now the equation becomes: \(u^2 - 5u + 6 = 0\).
1. Factorise: \((u-2)(u-3) = 0\).
2. Solve for u: \(u = 2\) or \(u = 3\).
3. Switch back to x: \(e^x = 2\) or \(e^x = 3\).
4. Find x: \(x = \ln(2)\) or \(x = \ln(3)\).

Quick Review Box:
- If you see a term with \(2x\) in the power and another with \(x\), think Quadratic!
- Warning: If you get a negative value for \(u\) (e.g., \(e^x = -2\)), there is no solution for that part, because \(e^x\) is always positive!

Summary: Your Exponential Toolkit

1. Check for Same Bases: Can you write both sides as \(2^n\) or \(3^n\)?
2. Use Logs for Different Bases: Take \(\ln\) of both sides to bring the variable down.
3. Use Substitution for Quadratics: Look for the \(e^{2x}\) and \(e^x\) pattern.
4. Check Your Answers: Plug your decimal value back into the original equation to see if it works!

Don't worry if this feels like a lot of steps! Like any sport or game, you just need to practice the "moves" until they become second nature. You've got this!