Equations involving exponentials

Introduction: Unlocking the Exponent

Welcome to one of the most useful chapters in A Level Maths! Have you ever wondered how scientists predict how quickly a virus spreads, or how a bank calculates interest on your savings? They use exponential equations. In this chapter, we are going to learn how to solve equations where the unknown value (usually \(x\)) is "trapped" up in the exponent (the power).

Don't worry if this seems a bit "alien" at first. By the end of these notes, you'll have a toolkit of tricks—like using logarithms—to bring those variables down to earth and solve them with ease.

The Basics: Prerequisites You Need

Before we dive in, let’s quickly refresh two things you already know:

1. Laws of Indices: Remember that \(a^m \times a^n = a^{m+n}\) and \((a^m)^n = a^{mn}\). These rules are the "logic" behind everything we do here.
2. The "Log" Connection: Logarithms are the inverse of exponentials. If \(a^x = b\), then \(x = \log_a(b)\). Think of the logarithm as the "key" that unlocks the exponent.

Type 1: The "Same Base" Method

Sometimes, we get lucky. If both sides of an equation can be written using the same base, we can simply "cancel" the bases and set the powers equal to each other.

Example: Solve \(2^{x+1} = 8\)
1. Recognize that \(8\) is actually \(2^3\).
2. Rewrite the equation: \(2^{x+1} = 2^3\).
3. Since the bases are the same, the powers must be equal: \(x+1 = 3\).
4. Solve for \(x\): \(x = 2\).

Quick Review: If \(a^f(x) = a^g(x)\), then \(f(x) = g(x)\). Easy!

Type 2: Using Logarithms for \(a^x = b\)

What if the bases aren't the same? For example, \(3^x = 20\). You can't write \(20\) as a nice power of \(3\). This is where we take logs of both sides.

Step-by-Step Process:

1. Take the Natural Log (\(\ln\)): Apply \(\ln\) to both sides: \(\ln(3^x) = \ln(20)\).
2. The Power Rule: Use the law \(\log(a^k) = k\log(a)\) to bring the \(x\) down to the front: \(x\ln(3) = \ln(20)\).
3. Isolate \(x\): Divide both sides by \(\ln(3)\): \(x = \frac{\ln(20)}{\ln(3)}\).
4. Calculate: Plug it into your calculator: \(x \approx 2.73\).

Memory Aid: Think of the logarithm as a "gravity field" that pulls the exponent down to the main line where you can deal with it.

Did you know?

We usually use \(\ln\) (base \(e\)) because it’s the standard in A Level Maths, but you could use \(\log_{10}\) and get the exact same answer! Just be consistent on both sides.

Type 3: Equations with Different Bases on Both Sides

The syllabus (Ref 1.06g) specifically mentions equations like \(2^x = 3^{2x-1}\). This looks scary because there are \(x\) values on both sides with different bases. Don't panic! The "Log Trick" still works.

Example: Solve \(2^x = 3^{2x-1}\)
1. Take logs: \(\ln(2^x) = \ln(3^{2x-1})\).
2. Bring down the powers: \(x\ln(2) = (2x-1)\ln(3)\).
3. Expand the brackets: \(x\ln(2) = 2x\ln(3) - \ln(3)\).
4. Get all \(x\) terms on one side: \(\ln(3) = 2x\ln(3) - x\ln(2)\).
5. Factorize \(x\): \(\ln(3) = x(2\ln(3) - \ln(2))\).
6. Divide to find \(x\): \(x = \frac{\ln(3)}{2\ln(3) - \ln(2)}\).

Key Takeaway: When you have multiple \(x\) terms, treat \(\ln(2)\) or \(\ln(3)\) just like a normal number (like \(5\) or \(10\)). Expand, collect, and factorize!

Type 4: "Hidden" Quadratics

Sometimes an exponential equation is actually a quadratic equation in disguise. These usually involve terms like \(e^{2x}\) and \(e^x\).

Remember from your index laws: \(e^{2x}\) is the same as \((e^x)^2\).

Example: Solve \(e^{2x} - 5e^x + 6 = 0\)
1. Substitution: Let \(u = e^x\).
2. Rewrite: The equation becomes \(u^2 - 5u + 6 = 0\).
3. Factorize: \((u - 2)(u - 3) = 0\).
4. Solve for \(u\): \(u = 2\) or \(u = 3\).
5. Solve for \(x\): Switch back to \(e^x\). So, \(e^x = 2\) or \(e^x = 3\).
6. Final Answer: \(x = \ln(2)\) or \(x = \ln(3)\).

Common Mistake to Avoid: Exponentials like \(e^x\) or \(2^x\) can never be negative or zero. If you solved the quadratic and got \(u = -5\), you would write "no solution" for that part, because \(e^x = -5\) is impossible!

Common Pitfalls & How to Avoid Them

1. Splitting Logs Incorrectly: Remember \(\ln(A + B)\) is NOT \(\ln(A) + \ln(B)\). You can only use the power rule on single log terms.
2. The Brackets Trap: In Type 3, when you bring down a power like \((2x-1)\), always put it in brackets! Failing to multiply the whole power by the log is the #1 way students lose marks.
3. Calculator Errors: When typing \(x = \frac{\ln(3)}{2\ln(3) - \ln(2)}\), use the fraction button on your calculator to ensure the denominator is grouped correctly.

Summary: Your "Solving Exponentials" Checklist

Is it a simple base match? If yes, equate the powers.
Is it a "single" exponential \(a^x = b\)? Take logs and use the power rule.
Are there different bases on both sides? Take logs, expand, and factorize \(x\).
Does it look like a quadratic? Use substitution (\(u = a^x\)), solve, and convert back.

Key Takeaway: Logarithms are your best friend. Whenever \(x\) is in the air, use a log to bring it down!