They agree twice and then part company

2 to the x against 2x x = 3.0

8.002 to the x

6.002x

1.33 timesratio

Two agreements, at x = 1 and x = 2, and nothing else. A curve and a line can cross twice without being alike anywhere.

The figure needs JavaScript. It plots y = 2^x against y = 2x: both are 2 at x = 1 and 4 at x = 2, then 8 against 6, 16 against 8, and 1024 against 20 by x = 10. A second view draws y = 2^x and y = log₂x as reflections in y = x, through (0, 1) and (1, 0). Everything below stands on its own.

The two families

	Exponential	Logarithmic
Written	f(x) = a^x, a > 0, and e^x	f(x) = log_ax, with x > 0, a > 0 and a ≠ 1, and ln x
Domain	every real x	x > 0 only
Range	y > 0 only	every real y
Always through	(0, 1)	(1, 0)
Asymptote	y = 0, the x-axis	x = 0, the y-axis

The a ≠ 1 is not fussiness: 1^x is the constant 1, so it has no inverse and log₁ does not exist. Read the table across and the symmetry is obvious: every row swaps. Domain and range swap, the two special points swap, and the asymptote swaps axes, because the two functions are inverses and an inverse is a reflection in y = x.

Why one base is enough

Every exponential can be written with base e:

a^x = e^{x ln a}

because taking the natural log of both sides gives x ln a either way. Check it: 2³ = 8, and e^{3 ln 2} = e^2.079 = 8.

And the two operations undo each other exactly:

log_a(a^x) = x and a^log_ax = x

so log₂(2⁵) = 5 without computing 32 first. That is what "inverse" means here, and it is the fastest simplification in the topic.

ln 2 = 0.693, which is the number everything about doubling turns on.

An exponential is not a steep straight line. It is a curve whose rate grows with its size, and that is a different thing entirely. 2^x adds 1 going from x = 0 to x = 1, and adds 512 going from x = 9 to x = 10. A straight line adds the same amount every step, for ever. So "exponential" is not a word for "fast"; it is a word for "the step is proportional to the size". When a > 1 that means the step grows, which is the case above. When 0 < a < 1 it means the step shrinks, which is radioactive decay and a discharging capacitor, and those are exponential too. That is the Aim 8 point the guide raises about the popular use of the term.

Reflections in y = x

Because they are inverses, the graph of log_ax is the graph of a^x reflected in y = x. Every feature follows:

2³ = 8, so log₂8 = 3. The point (3, 8) on one becomes (8, 3) on the other.
a⁰ = 1 for every a, so every exponential passes through (0, 1); reflected, every logarithm passes through (1, 0).
e¹ = 2.718 and ln e = 1. One point and its reflection.
ln 1 = 0, and no logarithm has a value at zero or below, because no power of a positive base is zero or negative.

A sketch of either curve is a sketch of both, if you draw y = x as well. That is worth a mark on any question asking for the inverse.

Three things that follow from the domain, and cost marks.

ln(−3) does not exist. If an equation gives it, that root is rejected, and saying so is the mark.
a^x is never zero, so an equation like e^x = 0 has no solutions at all, and e^x = −2 likewise.
The base must be positive, so a^x with a = −2 is not one of these functions and does not have a graph of this kind.

On the GDC: the two curves and their reflection

Drawing an exponential and its logarithm with y = x between them makes the inverse relationship something you see rather than something you are told, and it takes three entries.

When you may use it. Analysis Paper 1 is non-calculator, and sketching these pairs with their asymptotes and special points is Paper 1 work. Paper 2 is where a numerical solution to an exponential equation is wanted.

TI-Nspire CX II

A Graphs page with f1(x)=2^x, f2(x)=log(x,2) and f3(x)=x Use the x² key or the right arrow to leave the exponent: typing ^ opens a superscript box and everything after it stays inside.
menu → Window / Zoom → Zoom Square, or the reflection will not look like one
To see the agreement in this page's claim, add f4(x)=2x and use Analyze Graph → Intersection twice: 1 and 2
On a Calculator page, e^(3*ln(2)) returns 8, which is the base-change relationship in one line

Casio fx-CG50

MENU → Graph with Y1=2^X, Y2=logab(2,X) and Y3=X
SHIFT F3 V-Window → F1 INIT gives the square-ish default window the reflection needs. F2 is TRIG, which is not square and will make the reflection look wrong
The logab command is OPTN → F4 CALC, and the BASE comes first
SHIFT F5 G-SOLVE → ISCT, with the right arrow for the second crossing

The mark people lose. Writing 2x where 2^x belongs, or the reverse. They agree at exactly two points and a question is unlikely to use either of them. The habit: say the expression out loud. "Two to the x" and "two x" are different words, and students who only ever read them silently blur them under pressure. On the Casio, note also that logab wants the base first, so logab(2,X) is log base 2 of x and the other order gives something else without complaining.

Your turn

1. Find 2¹⁰.

2. And 2 × 10.

3. Find log₂8.

4. Find ln 2, to 3 decimal places.

5. Why does every exponential graph pass through (0, 1)?

Question 5. Why does every exponential graph pass through the point 0, 1? Because the graphs are all drawn from the same template. Because a to the power 0 is 1 for every positive base a. Because the asymptote is y = 0. They do not all; it depends on the base.

Where the marks go

1 markThe right shape, with the asymptote drawn.

1 markThe special point, (0, 1) or (1, 0), marked.

1 markThe domain or range, where asked.

On a question asking for the inverse, drawing y = x and reflecting is accepted working and is quicker than rearranging. Label the line, or the reflection is just two curves.

The two families

Why one base is enough

Reflections in y = x

On the GDC: the two curves and their reflection

TI-Nspire CX II

Casio fx-CG50

Your turn

Want a verdict on your own draft?