• Exponential growth/decay means a quantity changes by multiplying by the same scale factor every time period (not adding a fixed amount).
• The model is y = a^x. If a > 1 → growth. If 0 < a < 1 → decay.
• Every graph of y = a^x passes through (0, 1) and has a horizontal asymptote at y = 0 — it never actually reaches zero.
• Adding a coefficient, y = ba^x, shifts the y-intercept to (0, b) — b is your starting amount.
• A logarithm is the inverse of an exponential — it answers "what power do I need?"
• log_a(b) = x means exactly the same thing as a^x = b. They're two ways of writing one fact.
• No base written (like log(50)) always means base 10.
• You can solve a^x = b for x using logs: x = log_a(b), or with the base-10 shortcut x = log(b) / log(a).

Here's the key distinction that trips people up: linear change adds or subtracts the same amount every step (like +£5 every week). Exponential change multiplies by the same factor every step (like ×1.05 every week — 5% more each time).

That's why exponential change feels "slow, then suddenly explosive" — each new increase is bigger than the last, because you're multiplying an ever-growing number by the same factor, not adding a fixed chunk to it.

Real-life growth examples: population increase, bacteria multiplying, viral infection spread, compound interest.

Real-life decay examples: a hot drink cooling down, a car's value depreciating, radioactive decay, medicine leaving your bloodstream.

• If a > 1, you're multiplying by something bigger than 1 each time → the value grows.
• If 0 < a < 1 (a fraction/decimal less than 1), you're multiplying by something smaller than 1 each time → the value shrinks toward zero.
• a can never be negative in these models — a negative base would make the graph flip signs every step, which doesn't make physical sense for population, mass, temperature, etc.

Every graph of y = a^x shares two fixed features, no matter what a is:

• It always passes through (0, 1). Why? Because anything to the power 0 is 1: a⁰ = 1.
• It has a horizontal asymptote at y = 0. The curve gets closer and closer to the x-axis but never touches it, because there's no power you can raise a positive number to that gives exactly 0.

   y

   |            y = aˣ  (a > 1, growth)

   |                                  ╱

   |                              ╱

   |                          ╱

   |  y = aˣ (0

    
The graph never crosses y = 0. If a question shows an exponential curve touching or crossing the x-axis, something's wrong — go back and check your working.



    


      What if there's a coefficient b? For y = bax, the "1" gets replaced by "b" — since y = ba0 = b(1) = b at x = 0. So b is simply the starting amount before any growth/decay has happened, and it stretches the whole graph vertically by a factor of b.

    

  

A logarithm is the inverse function of an exponential. If an exponential asks "what do I get if I raise a to the power x?", a logarithm asks the reverse question: "what power do I need to raise a to, in order to get b?"

Reading it out loud: "log base a of b" is pronounced exactly like it looks — it's asking "what power do I raise a to, to get b?"

Let's build up a feel for this with base 5, working out what power gets us to different numbers:

Notice the pattern: negative logs come from fractions less than 1, and fractional logs come from roots. This isn't a coincidence — it's the same index laws you already know, just viewed from the other direction.

This is the single most useful skill in this whole chapter — being able to flip fluently between the two forms. Here's the trick that makes it click:

Some exponential equations solve "by inspection" — you can just see the answer. For example, 2^x = 8 is obviously x = 3, because you know 2³ = 8. But what about 2^x = 10? There's no whole number that works — this is where logs become essential.

A classic exam scenario: you know the growth rate and the target value, and you need to find how many time periods (n) it takes to get there. This is exactly the situation logs were built for.

Since logs undo exponentials, they're exactly the tool you need whenever you're asked to find the inverse of an exponential function.

• I can explain the difference between linear and exponential change
• I know when a > 1 means growth and 0 < a < 1 means decay
• I can state and explain why every y = aˣ graph passes through (0,1)
• I understand why the asymptote is at y = 0 and what that means physically
• I can explain what the coefficient b does in y = baˣ
• I can convert fluently between log form and exponential form
• I can evaluate simple logs (including negative and fractional results) by hand
• I know that "log" with no base means base 10
• I can solve aˣ = b using either log_a(b) or log(b)/log(a)
• I can apply logs to compound interest "how many years" problems, including rounding correctly
• I can find the inverse of an exponential function using logs
• I can rearrange equations with logs to make a variable the subject