Why can the base of an exponential function not be 1?

Because 1 to any power is 1, so the graph would be a horizontal line — a constant function, not an exponential one. The base must also be positive: for a negative base, say (−2)^(1/2), the result stops existing among the real numbers, so the domain would no longer be all of R.

When does an exponential function rise, and when does it fall?

It rises for a base greater than 1 and falls for a base between 0 and 1. That single parameter decides it: y = 3ˣ rises, y = (1/3)ˣ falls, and both share the horizontal asymptote y = 0 and both pass through (0, 1).

What is the range of an exponential function?

Every positive number, that is the interval (0, ∞). A power of a positive base is never zero and never negative — however negative the exponent, the result only shrinks towards zero. That is why the line y = 0 is a horizontal asymptote rather than a value that is taken.

How do you solve an exponential equation?

Bring both sides to the same base and compare the exponents. In 2^(x+1) = 32 write 32 as 2⁵, so x + 1 = 5 and x = 4. This is allowed because the exponential function is one-to-one: two equal powers of the same base must have equal exponents.

How do you solve a logarithmic equation?

Use the definition of a logarithm: log_b(A) = k means exactly A = b^k. So log₂(3x − 1) = 4 turns into 3x − 1 = 16, giving x = 17/3. At the end you must check the assumption: the argument has to be positive, and 3·(17/3) − 1 = 16 > 0, so the solution stands.

What is the domain of a logarithmic function?

Positive numbers only. A logarithm answers "to what power must the base be raised", and a power of a positive base always comes out positive — so there is no exponent giving zero or a negative number. Inside a composition the domain comes from an inequality: for f(x) = log₂(2x − 6) you need 2x − 6 > 0, that is x > 3.

Why are the graphs of y = bˣ and y = log_b x mirror images?

Because they are mutually inverse functions: each undoes what the other does. If the point (a, c) lies on the exponential graph, then (c, a) lies on the logarithmic one — and swapping the coordinates is precisely a reflection in the line y = x. Their asymptotes swap roles for the same reason: the horizontal one becomes vertical.

How does exponential growth differ from linear growth?

Linear growth adds the same amount at every step; exponential growth multiplies by the same factor. After ten steps a sequence adding 2 reaches 21 from 1, while one doubling reaches 1024. That is why an exponential function sooner or later outgrows every line and every polynomial, however slowly it starts.

Advanced level

Exponential and logarithmic functions

Grain on a chessboard, a loan, radioactive decay and the decibel scale — one pair of functions describes them all. See what the graph of y = bˣ looks like and why y = log_b x is its mirror image, how solving an exponential equation comes down to comparing exponents, and where the domain of a logarithm comes from.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Where this is used

Real situations where you count exactly the way this lesson teaches:

Carbon-14 dating
The half-life of carbon-14 is 5730 years, so N(t) = N₀·(1/2)^(t/5730). An archaeologist measures 25% of the original isotope content in a find and solves (1/2)^(t/5730) = 0.25, that is t/5730 = 2 — the sample is 11,460 years old. At 12.5% it would come out at 17,190 years: each further halving is one more full period.
Noise level in decibels
The decibel scale is logarithmic: L = 10·log₁₀(I/I₀). A conversation with an intensity a billion times the hearing threshold measures 90 dB, and two such conversations at once measure not 180 dB but 93 dB, because doubling the intensity adds 10·log₁₀2 ≈ 3 dB. An acoustician designing a hall computes exactly this logarithm; the [decibel converter](/en/decibels) handles the quick conversions.
Continuous compounding at a bank
A deposit of 20,000 zloty at 4% a year with continuous compounding grows as K(t) = 20000·e^(0.04t). After 5 years that is 24,428 zloty, and the question "when will it grow by half" is the equation e^(0.04t) = 1.5, so t = ln 1.5 / 0.04 ≈ 10.1 years. Without a logarithm that answer cannot be computed at all, only guessed at.
The pH of a solution
pH = −log₁₀[H⁺], so a solution with a hydrogen-ion concentration of 10⁻³ mol/l has pH 3. Orange juice at pH 3.5 is exactly 10 times less acidic than vinegar at pH 2.5, although only one step separates them on the scale — a food technologist reads a factor out of that single step, not a difference.

All formulas

The exponential function
$f(x) = b^x, \qquad b > 0,\ b \neq 1$
domain: \mathbb{R}, range: (0,\ \infty)
The logarithmic function
$f(x) = \log_b x, \qquad b > 0,\ b \neq 1$
domain: (0,\ \infty), range: \mathbb{R}
Mutually inverse
$b^{\log_b x} = x, \qquad \log_b b^x = x$
each one undoes what the other does
Exponential equation
$b^{x_1} = b^{x_2} \iff x_1 = x_2$
one-to-one — which is why the exponents may be compared
Logarithmic equation
$\log_b x_1 = \log_b x_2 \iff x_1 = x_2$
given x_1 > 0 and x_2 > 0
Radioactive decay
$N(t) = N_0 \cdot \left(\tfrac{1}{2}\right)^{t/T}$
T — the half-life

The legend of the inventor of chess says he asked for grain: one on the first square, two on the second, four on the third and so on. The sixty-fourth square carries $2^{63}$ grains — more than humanity has produced in its entire history. That is not an exaggeration but an exponential function.

The exponential function

f(x) = b^x, \qquad b > 0, \; b \neq 1

The variable stands in the exponent, not in the base — which is what separates $2^x$ from $x^2$ . The conditions on the base are not whims:

$b > 0$ , because for a negative base a rational exponent stops existing among the reals — $(-4)^{1/2}$ has no value;
$b \neq 1$ , because $1^x = 1$ for every $x$ , so the graph would be a horizontal line.

Growth and decay. Both curves pass through (0, 1), because b⁰ = 1, and both have the horizontal asymptote y = 0 — the values shrink towards zero without ever being zero.

property	$b > 1$	$0 < b < 1$
domain	$\mathbb{R}$	$\mathbb{R}$
range	$(0,\, \infty)$	$(0,\, \infty)$
monotonicity	increasing	decreasing
common point	$(0,\, 1)$	$(0,\, 1)$
asymptote	$y = 0$	$y = 0$

The range is the positive numbers and nothing else. A power of a positive base is never zero and never negative — for very small exponents it only shrinks, approaching the horizontal asymptote $y = 0$ .

Shifts

The formula $y = a \cdot b^{x-p} + q$ reads like any other graph transformation: a translation by $[p,\, q]$ . One consequence is worth remembering: the asymptote travels with the curve, so after the shift it sits at $y = q$ rather than at $y = 0$ .

The logarithmic function

A logarithm is the operation inverse to raising to a power — $\log_b x$ answers "to what power must $b$ be raised to give $x$ ". Written as a function:

f(x) = \log_b x, \qquad b > 0, \; b \neq 1

The domain is the positive numbers only, and this follows straight from the paragraph above: since a power of a positive base is always positive, there is no exponent that would give $0$ or a negative number. So there is nothing there to take a logarithm of.

A mutually inverse pair: y = 2ˣ and y = log₂ x, reflected in the line y = x. The horizontal asymptote of one becomes the vertical asymptote of the other, and the point (0, 1) goes to (1, 0).

The two functions are mutually inverse, which we write as two identities:

b^{\log_b x} = x \quad (x > 0), \qquad \log_b b^{\,x} = x

A reflection in the line $y = x$ swaps the coordinates — and that is the whole content of being inverse, treated at length in composition and inverse functions. The asymptotes and the two sets swap roles as well:

	$y = b^x$	$y = \log_b x$
domain	$\mathbb{R}$	$(0,\, \infty)$
range	$(0,\, \infty)$	$\mathbb{R}$
special point	$(0,\, 1)$	$(1,\, 0)$
asymptote	horizontal $y = 0$	vertical $x = 0$

Exponential equations

The core curriculum solves these through the properties of the functions rather than with an apparatus of their own. The key property is being one-to-one:

b^{x_1} = b^{x_2} \iff x_1 = x_2

Two equal powers of the same base must have equal exponents — because an exponential function is strictly monotone, so two different arguments cannot produce the same value. The whole method is therefore: bring both sides to the same base and compare the exponents.

Solve the equation 3^(2x − 1) = 81.

When a common base cannot be found, you take a logarithm of both sides — and that is exactly the moment the pair of functions starts working together.

Logarithmic equations

Here the key is the definition of a logarithm itself:

\log_b A = k \iff A = b^k

together with — just as above — the one-to-one property, which turns $\log_b x_1 = \log_b x_2$ into $x_1 = x_2$ .

Domain assumptions are written down before solving, not after. The argument of a logarithm must be positive, so the work starts with an inequality.

Solve the equation log₂(3x + 2) = 4.

The domain is sometimes the question in its own right. For $f(x) = \log_3(2x - 8)$ we need $2x - 8 > 0$ , that is $x > 4$ : the domain is the interval $(4,\, \infty)$ . Mind the sign — a negative coefficient flips the inequality.

Why this is everywhere

Exponential growth turns up wherever the increase is proportional to the amount present: money under compound interest, a bacterial population, the number of cases early in an epidemic. Exponential decay turns up wherever the loss is proportional to what is left: radioactive decay, coffee cooling, a drug clearing from the blood.

N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{t/T}

where $T$ is the half-life — the time after which half remains. After two such periods a quarter is left, after three an eighth.

Logarithms, in turn, serve to compress a scale: decibels, pH, earthquake magnitude and stellar magnitude are all logarithmic scales. Thanks to them a thousandfold difference in intensity fits into three marks instead of a thousand.

Practice

The set checks three skills: an exponential equation (bring to a common base, compare exponents), a logarithmic equation (apply the definition, then check the assumption) and the domain of a logarithm, which you type as an inequality, e.g. x > 3. Evaluating a single power or logarithm is practised in the numbers branch — here the subject is the functions, not the individual operation.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

3^(x − 2) = 9

Common mistakes

Confusing $2^x$ with $x^2$ — in the first the variable is the exponent, in the second the base, and they grow in completely different ways.
Hunting for a zero of an exponential function — there is none; $b^x$ is always positive, and $y = 0$ is an asymptote.
Taking a logarithm of zero or of a negative number — the domain of a logarithm is the positive numbers only.
Skipping the assumptions in a logarithmic equation — the arithmetic can produce a number outside the domain, and it must then be rejected.
Flipping the inequality for a positive coefficient — in $\log_2(3x - 6)$ the condition is $x > 2$ , not $x < 2$ ; the sign flips only when dividing by a negative number.
Believing $y = 2^{x} + 3$ has the asymptote $y = 0$ — the upward shift carries the asymptote with it, so it sits at $y = 3$ .

Formula card

Topic: Exponential and logarithmic functions

The exponential function
$f(x) = b^x, \qquad b > 0,\ b \neq 1$
domain: \mathbb{R}, range: (0,\ \infty)
The logarithmic function
$f(x) = \log_b x, \qquad b > 0,\ b \neq 1$
domain: (0,\ \infty), range: \mathbb{R}
Mutually inverse
$b^{\log_b x} = x, \qquad \log_b b^x = x$
each one undoes what the other does
Exponential equation
$b^{x_1} = b^{x_2} \iff x_1 = x_2$
one-to-one — which is why the exponents may be compared
Logarithmic equation
$\log_b x_1 = \log_b x_2 \iff x_1 = x_2$
given x_1 > 0 and x_2 > 0
Radioactive decay
$N(t) = N_0 \cdot \left(\tfrac{1}{2}\right)^{t/T}$
T — the half-life

The exponential function y = 2ˣ and the logarithmic function y = log₂ x. Each is the reflection of the other in the line y = x: what one computes, the other undoes.

Growth and decay: y = 2ˣ rises, y = (1/2)ˣ falls. Both curves share the horizontal asymptote y = 0 and both pass through (0, 1), because any base to the power zero is one.

Frequently asked questions

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min

Exponential and logarithmic functions

Before you start

Where this is used

All formulas

The exponential function

Shifts

The logarithmic function

Exponential equations

Logarithmic equations

Why this is everywhere

Practice

Practice

Common mistakes

Formula card

Frequently asked questions

Why can the base of an exponential function not be 1?

When does an exponential function rise, and when does it fall?

What is the range of an exponential function?

How do you solve an exponential equation?

How do you solve a logarithmic equation?

What is the domain of a logarithmic function?

Why are the graphs of y = bˣ and y = log_b x mirror images?

How does exponential growth differ from linear growth?

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Why a function's graph can never double back