Exponential and logarithmic functions are inverses of each other — every feature of one is the mirror image of the other in the line y = x. Learn one, you've nearly learned the other.
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What you need to know
Exponential:f(x) = ax with a > 0 — domain ℝ, range positive reals
Natural exponential:f(x) = ex where e ≈ 2.718 (Euler's constant)
Logarithmic:f(x) = logax with x > 0 — domain positive reals, range ℝ
Natural log: ln x ≡ logex — the inverse of ex
Inverse rule:y = ax ⟺ x = logay
Exponential graphs always pass through (0, 1); logarithmic graphs through (1, 0)
Both have one asymptote: exponential at y = 0, logarithmic at x = 0
Exponential Functions
An exponential function has the form:
f(x) = ax, a > 0
The domain is all real numbers (you can put any x in), and the range is all positive real numbers (the output is always > 0). The most important exponential function is f(x) = ex, where e ≈ 2.718 is Euler's constant. Any other exponential can be rewritten in terms of e:
ax = ex ln a(formula booklet)
The Two Cases
The shape depends on whether a is greater than or less than 1:
a > 1 (Growth)
Increases. As x → −∞, y → 0. As x → ∞, y → ∞.
0 < a < 1 (Decay)
Decreases. As x → −∞, y → ∞. As x → ∞, y → 0.
Key Features of y = ax
y-intercept at (0, 1) — every exponential passes through here, since a0 = 1
Always passes through (1, a) — useful checkpoint
No roots — the graph never touches the x-axis
Horizontal asymptote at y = 0
No turning points — exponentials only ever go one way
Logarithmic Functions
A logarithmic function has the form:
f(x) = logax, x > 0
The domain is positive real numbers only (you can't take the log of zero or a negative). The range is all real numbers. The most important logarithmic function is f(x) = ln x — the natural log, which is just logex. Any other logarithm can be converted using the change of base formula:
logax = ln xln a
The Logarithmic Graph y = logax (a > 1)
Key Features of y = logax
No y-intercept — the curve never crosses the y-axis
One root at (1, 0) — every logarithm passes through here, since loga1 = 0
Always passes through (a, 1)
Vertical asymptote at x = 0
No turning points
The Inverse Relationship
This is the big idea: exponentials and logarithms undo each other. They're inverse functions, which means:
ln(ex) = x and eln x = x
And graphically, the curves are reflections of each other in the line y = x:
Reflection in y = x
Notice the symmetry:ex passes through (0, 1) — and ln x passes through (1, 0). The coordinates are swapped. That's exactly what reflection in y = x does.
Side-by-Side Comparison
Every feature of exponentials becomes the swapped feature of logarithms:
e
Exponential
y = ax
Domain:x ∈ ℝ
Range:y > 0
y-intercept: (0, 1)
No x-intercept
Passes through (1, a)
Horizontal asymptote at y = 0
log
Logarithmic
y = logax
Domain:x > 0
Range:y ∈ ℝ
No y-intercept
x-intercept: (1, 0)
Passes through (a, 1)
Vertical asymptote at x = 0
Worked Examples
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Example 1 — Identify features of an exponential
For f(x) = 3x, state: (a) the y-intercept, (b) a known point on the curve, (c) the equation of the asymptote.
Answer:
(a) Every exponential passes through (0, 1).(0, 1)(b) Always passes through (1, a).a = 3, so (1, 3)(1, 3)(c) Horizontal asymptote.y = 0
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Example 2 — Convert an exponential to base e
Express 4x in the form ekx, giving the exact value of k.
Answer:
Use the formula ax = ex ln a.4x = ex ln 4Compare with ekx:k = ln 4Or equivalently k = 2 ln 2 (since ln 4 = ln 2² = 2 ln 2).
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Example 3 — Find the inverse of a logarithm
The function f is defined by f(x) = log5x for x > 0.
(a) Write down the inverse of f. Give your answer in the form eg(x).
Step 1: swap x and y in y = log₅ x.x = log₅ y ⟺ y = 5xStep 2: convert 5x to base e using ax = ex ln a.5x = ex ln 5f−1(x) = ex ln 5
(b) Sketch f and its inverse on the same set of axes.
Sketch y = log₅ x:passes through (1, 0) and (5, 1), vertical asymptote x = 0Sketch its inverse y = 5x = ex ln 5:passes through (0, 1) and (1, 5), horizontal asymptote y = 0The two curves are reflections in y = x.Always include the line y = x as a dashed reference line on the sketch.
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Example 4 — Use the inverse relationship
Solve for x: (a) ln(e3x) = 12(b) eln(2x + 1) = 7.
Answer:
(a) Use ln(ex) = x.ln(e3x) = 3x = 12x = 4(b) Use eln x = x.eln(2x + 1) = 2x + 1 = 72x = 6x = 3Inverse functions cancel each other out — that's the whole point.
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Example 5 — Change of base
Use the change of base formula to evaluate log3 50, giving your answer to 3 significant figures.
Answer:
Apply loga x = ln x / ln a.log₃ 50 = ln 50 / ln 3Compute on a calculator.= 3.9120... / 1.0986...= 3.5608...log₃ 50 ≈ 3.56 (3 s.f.)Most calculators only have ln and log (base 10). Change of base is essential for any other base.
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Tips
Memorise the two anchor points. Exponential always passes through (0, 1) and (1, a). Logarithm always passes through (1, 0) and (a, 1). The pairs are swapped — that's the inverse relationship in action.
Convert any exponential to base e using ax = ex ln a. This is in the formula booklet — use it!
Convert any log to ln using logax = ln xln a. Calculators only natively have ln and log10.
Sketch with the line y = x when showing inverses on one set of axes — it makes the reflection visible.
Use the GDC to verify graph shape, but remember most GDCs don't draw asymptotes — add them by hand.
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Common mistakes
Forgetting the domain restriction on logarithms. x > 0 always — you can't take the log of zero or a negative.
Mixing up which axis is the asymptote. Exponential → horizontal (y = 0). Logarithm → vertical (x = 0).
Drawing the curve crossing the asymptote. Asymptotes are barriers — the curve approaches but never touches them.
Writing logax = ln aln x (upside down). The base goes on the bottom, not the top.
Confusing a > 1 (growth) with 0 < a < 1 (decay). Always check the value of a before sketching.
Treating ex as a power, not as a function.ex · ey = ex+y, not exy.
Final word: Exponentials and logs are inverse functions, so the features of one are the features of the other with x and y swapped. Memorise the anchor points (0, 1), (1, a), and (1, 0), (a, 1) — and the conversions to base e and ln. Everything else flows from those.
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