Logarithms as inverses of exponentials

Intuition [Beginner]

A logarithm answers one question: "what power do I need?" If $2^3=8$, then the logarithm base $2$ of $8$ is $3$. You read ${\log }_{2}8=3$ as "$2$ to the power $3$ gives $8$."

The logarithm reverses the exponential. The exponential asks "what do I get when I raise $a$ to this power?" The logarithm asks "what power of $a$ gives me this number?" They are two sides of the same coin, like multiplication and division. One builds up; the other takes apart.

Why does this matter? Because logarithms turn multiplication into addition. Computing $128\times 256$ by hand is work. But ${\log }_{2}128=7$ and ${\log }_{2}256=8$, and $7+8=15$, and $2^{15}=32768$. The product $128\times 256=32768$. You replaced multiplication with addition by passing through the logarithm. This is the reason logarithms were invented.

Visual [Beginner]

The graph of $y={\log }_{2}x$ is the mirror image of $y=2^x$, reflected across the diagonal line $y=x$. Where $2^x$ climbs steeply to the right, ${\log }_{2}x$ climbs slowly. Where $2^x$ passes through $(0,1)$, ${\log }_{2}x$ passes through $(1,0)$.

The logarithm is defined only for positive inputs. The curve rises without bound as $x$ grows, but it rises slowly: reaching $y=10$ requires $x=2^{10}=1024$. As $x$ approaches $0$ from the right, the logarithm plunges toward negative infinity, never touching the vertical axis.

Worked example [Beginner]

Compute ${\log }_{3}81$.

Step 1. Ask: "three to what power equals $81$?"

Step 2. Test small powers of $3$: $3^1=3$, $3^2=9$, $3^3=27$, $3^4=81$.

Step 3. The answer is $4$.

What this tells us: ${\log }_{3}81=4$. The logarithm found the exponent by asking the exponential to undo itself. You can always verify a logarithm answer by running the exponential forward: $3^4=81$ confirms ${\log }_{3}81=4$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $a>0$, $a\ne 1$. The logarithm base $a$ is the function ${\log }_a:(0,\infty )\to \mathbb{R}$ defined by:

$${\log }_{a}y=x⟺a^x=y.$$

This definition is well-posed because $f(x)=a^x$ is a strictly monotone continuous bijection from $\mathbb{R}$ onto $(0,\infty )$. Strict monotonicity guarantees injectivity (each output comes from at most one input), and the intermediate value theorem guarantees surjectivity onto $(0,\infty )$ because $a^x\to 0^{+}$ as $x\to -\infty$ and $a^x\to \infty$ as $x\to \infty$ when $a>1$ (the case $0<a<1$ reverses the limits). The inverse of a strictly monotone continuous function is itself strictly monotone and continuous.

The defining identities are the inverse relations:

$$a^{{\log }_{a}y}=y(y>0),{\log }_a(a^x)=x(x\in \mathbb{R}).$$

Counterexamples to common slips

• Base $a=1$ is excluded. The function $1^x=1$ is constant, so it has no inverse. The equation ${\log }_{1}y$ has no meaning because every $x$ satisfies $1^x=1$.
• The domain of ${\log }_a$ is $(0,\infty )$, not all of $\mathbb{R}$. Taking the logarithm of a negative number or zero is not defined within the reals because $a^x>0$ for all $x$.
• ${\log }_a(xy)$ is not ${\log }_{a}x\cdot {\log }_{a}y$. The correct law is ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$. The logarithm converts multiplication into addition, not into multiplication.

Key theorem with proof [Intermediate+]

Theorem (laws of logarithms). Let $a>0$, $a\ne 1$, and $x,y>0$, $p\in \mathbb{R}$. Then:

1. ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$.
2. ${\log }_a(x^p)=p\cdot {\log }_{a}x$.
3. ${\log }_a(x/y)={\log }_{a}x-{\log }_{a}y$.
4. (Change of base) ${\log }_{b}x=\frac{{\log }_{a}x}{{\log }_{a}b}$ for any base $b>0$, $b\ne 1$.

Proof of (1). Set $u={\log }_{a}x$ and $v={\log }_{a}y$, so $a^u=x$ and $a^v=y$. Then $xy=a^u\cdot a^v=a^{u+v}$ by the exponential addition law. Applying the definition of logarithm to both sides gives ${\log }_a(xy)=u+v={\log }_{a}x+{\log }_{a}y$. $\square$

Proof of (2). Set $u={\log }_{a}x$, so $a^u=x$. Then $x^p=(a^u{)}^p=a^{up}$ by the power-of-a-power law for exponentials. The definition of logarithm gives ${\log }_a(x^p)=up=p\cdot {\log }_{a}x$. $\square$

Proof of (3). Write $x/y=x\cdot y^{-1}$. By law (1), ${\log }_a(x\cdot y^{-1})={\log }_{a}x+{\log }_a(y^{-1})$. By law (2) with $p=-1$, ${\log }_a(y^{-1})=-{\log }_{a}y$. Combining: ${\log }_a(x/y)={\log }_{a}x-{\log }_{a}y$. $\square$

Proof of (4). Set $t={\log }_{b}x$, so $b^t=x$. Take ${\log }_a$ of both sides: ${\log }_a(b^t)={\log }_{a}x$. By law (2), $t\cdot {\log }_{a}b={\log }_{a}x$. Since ${\log }_{a}b\ne 0$ (because $a\ne 1$ and $b\ne 1$ imply $b$ is not a power of $a$ equal to $1$), divide: $t={\log }_{a}x/{\log }_{a}b$. $\square$

Bridge. The foundational reason logarithms convert multiplication into addition is that exponentials convert addition into multiplication — the logarithm is the inverse of that conversion, so it runs the process backwards. This is exactly the content of law (1): ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$ is the image under inversion of $a^{u+v}=a^u\cdot a^v$. The central insight is that every logarithm law is the mirror image of an exponential law, and the bridge is that the logarithm identifies the multiplicative group $({\mathbb{R}}_{>0},\times )$ with the additive group $(\mathbb{R},+)$ by inverting the exponential isomorphism. This pattern builds toward 02.06.01 where the natural logarithm is defined independently as an integral and the exponential is recovered as its inverse, and appears again in 00.02.05 where the general notion of inverse function is introduced. The change-of-base formula puts these together: it identifies every logarithm with a scalar multiple of every other, so all logarithm bases carry the same structural information.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (the natural logarithm). The base-$e$ logarithm $\ln x={\log }_{e}x$ is called the natural logarithm. For any base $a>0$, $a\ne 1$: ${\log }_{a}x=\ln x/\ln a$.

The natural logarithm is the unique logarithm whose base $e$ satisfies $d(e^x)/dx=e^x$. Equivalently, $\ln x$ is the inverse of the unique exponential whose derivative equals itself. This gives the natural logarithm a distinguished role in calculus and analysis.

Theorem 2 (logarithm as integral). The natural logarithm admits the integral representation:

$$\ln x={\int }_1^x\frac{1}{t}dt,x>0.$$

This integral defines $\ln x$ independently of the exponential: $\ln 1=0$, $\ln$ is strictly increasing and concave, and its inverse is $e^x$. The proof that this integral's inverse satisfies $(e^x{)}^{\prime }=e^x$ is the foundational link between the algebraic and analytic constructions.

Theorem 3 (concavity of logarithm). The function $\ln x$ is strictly concave on $(0,\infty )$: for all $x,y>0$ and $0<\lambda <1$:

$$\ln (\lambda x+(1-\lambda )y)>\lambda \ln x+(1-\lambda )\ln y.$$

This is the content of the AM-GM inequality in logarithmic form: $\ln$ being concave is equivalent to $(x+y)/2\ge \sqrt{xy}$ because applying $\ln$ to both sides gives $\ln ((x+y)/2)\ge (\ln x+\ln y)/2$, which reverses under exponentiation.

Theorem 4 (logarithmic growth is slower than any positive power). For every $\epsilon >0$:

$$\underset{x\to \infty }{\lim }\frac{\ln x}{x^{\epsilon }}=0.$$

This is the dual of the exponential growth theorem from 00.05.01. The proof uses the substitution $x=e^t$, giving $\ln (e^t)/e^{\epsilon t}=t/e^{\epsilon t}$, and the denominator grows exponentially while the numerator grows linearly.

Theorem 5 (logarithm and the prime-counting function). The function $\pi (n)$ counting primes $\le n$ satisfies $\pi (n)\sim n/\ln n$ as $n\to \infty$ (the prime number theorem, Hadamard 1896 and de la Vallee Poussin 1896). The natural logarithm enters through the Euler product $\zeta (s)={\prod }_p(1-p^{-s}{)}^{-1}$ and the pole at $s=1$, where the residue is controlled by $\ln$-growth.

Theorem 6 (logarithmic differentiation). For a product $f(x)=f_1(x)f_2(x)\cdots f_n(x)$ of differentiable positive functions, the derivative satisfies:

$$\frac{f^{\prime }(x)}{f(x)}=\frac{f_1^{\prime }(x)}{f_1(x)}+\frac{f_2^{\prime }(x)}{f_2(x)}+\cdots +\frac{f_n^{\prime }(x)}{f_n(x)}.$$

This is proved by writing $\ln f(x)=\ln f_1(x)+\cdots +\ln f_n(x)$ and differentiating both sides.

Theorem 7 (logarithmic scales and entropy). The Shannon entropy $H=-\sum p_i\ln p_i$ measures the information content of a probability distribution. The logarithm appears because additive information corresponds to multiplicative probability: two independent events with probabilities $p$ and $q$ give joint probability $pq$, and $\ln (pq)=\ln p+\ln q$ converts the product into a sum of contributions.

Synthesis. The foundational reason logarithms pervade mathematics is that they identify the multiplicative structure of positive reals with the additive structure of all reals. The central insight is that every property of the logarithm is an inverted property of the exponential: the product law for logarithms is the inverse of the addition law for exponentials, concavity of $\ln$ is the inverse of convexity of $\exp$, and the slow growth of $\ln$ is the inverse of the fast growth of $\exp$. This is exactly the content of the inverse-function relationship. The bridge is between algebra and analysis: putting these together, the logarithm converts multiplicative problems into additive ones (Theorem 7), measures growth rates through orders of magnitude (Theorems 4 and 5), and provides the differentiation shortcut for products (Theorem 6). The pattern generalises from real numbers to matrices via the matrix logarithm, to Lie groups via the exponential map, and to operator algebras via the spectral-theoretic functional calculus, each time carrying the same identification of multiplicative and additive structures.

Full proof set [Master]

Proposition 1 (the logarithm is well-defined). For $a>0$, $a\ne 1$, the map ${\log }_a:(0,\infty )\to \mathbb{R}$ is a well-defined function: for every $y>0$ there exists a unique $x\in \mathbb{R}$ with $a^x=y$.

Proof. If $a>1$, the function $f(x)=a^x$ is continuous, strictly increasing, ${\lim }_{x\to -\infty }{a}^x=0$, and ${\lim }_{x\to \infty }{a}^x=\infty$. By the intermediate value theorem, for each $y>0$ there exists $x$ with $a^x=y$. Strict monotonicity gives uniqueness. If $0<a<1$, the function $g(x)=a^x=(1/b{)}^x=b^{-x}$ where $b=1/a>1$ is strictly decreasing, continuous, and surjects onto $(0,\infty )$, so the same argument applies. $\square$

Proposition 2 (change of base). For $a,b>0$ with $a\ne 1$, $b\ne 1$, and any $x>0$: ${\log }_{b}x={\log }_{a}x/{\log }_{a}b$.

Proof. Set $t={\log }_{b}x$, so $b^t=x$. Take ${\log }_a$ of both sides: ${\log }_a(b^t)={\log }_{a}x$. By the power law for logarithms (Theorem 2 of the Key Theorem), this gives $t\cdot {\log }_{a}b={\log }_{a}x$. Since $b\ne 1$ and $a\ne 1$, ${\log }_{a}b\ne 0$. Solve: $t={\log }_{a}x/{\log }_{a}b$. $\square$

Proposition 3 (integral representation of $\ln$). The function $L(x)={\int }_1^{x}dt/t$ for $x>0$ satisfies the logarithm laws: $L(xy)=L(x)+L(y)$ and $L(x^p)=pL(x)$, and its inverse is an exponential function with base $e={\lim }_{n\to \infty }(1+1/n{)}^n$.

Proof. For the product law: $L(xy)={\int }_1^{xy}dt/t={\int }_1^{x}dt/t+{\int }_x^{xy}dt/t$. Substituting $t=xu$ in the second integral gives $dt=xdu$ and $dt/t=du/u$, with limits $u=1$ to $u=y$. So $L(xy)=L(x)+L(y)$. For the power law: $L(x^p)={\int }_1^{x^p}dt/t$. Substituting $t=u^p$ gives $dt=p{u}^{p-1}du$, so $dt/t=pdu/u$, with limits $u=1$ to $u=x$, giving $L(x^p)=p{\int }_1^{x}du/u=pL(x)$. The function $L$ is strictly increasing (its derivative $1/x>0$) and maps $(0,\infty )$ onto $\mathbb{R}$ (because $L(x)\to -\infty$ as $x\to 0^{+}$ and $L(x)\to \infty$ as $x\to \infty$). Its inverse is therefore a strictly increasing continuous bijection $\mathbb{R}\to (0,\infty )$ satisfying the exponential addition law, hence is $a^x$ for some $a>0$. Setting $a=L^{-1}(1)$ defines $e$: $L(e)=1$, so $e={\lim }_{n\to \infty }(1+1/n{)}^n$. $\square$

Connections [Master]

• Real exponents and the exponential function 00.05.01. The logarithm is the inverse of the exponential: every logarithm law is derived by inverting the corresponding exponential law. The product-to-sum rule ${\log }_a(xy)={\log }_{a}x+{\log }_{a}y$ is the mirror of $a^{u+v}=a^u\cdot a^v$, and the power rule ${\log }_a(x^p)=p{\log }_{a}x$ is the mirror of $(a^u{)}^p=a^{up}$. Without the exponential function defined in 00.05.01 as a continuous bijection from $\mathbb{R}$ onto $(0,\infty )$, the logarithm would have no definition.

• Functions, domain, and inverse 00.02.05. The logarithm is the precalc strand's central example of an inverse function. The exponential $f(x)=a^x$ is a bijection $\mathbb{R}\to (0,\infty )$, and ${\log }_a$ is its inverse $f^{-1}:(0,\infty )\to \mathbb{R}$. The domain restriction to $(0,\infty )$ for the logarithm's input is forced by the range of the exponential, providing a concrete instance of the general principle that the domain of an inverse function equals the range of the original.

• Real numbers and completeness 00.01.01. The existence of ${\log }_{a}y$ for every $y>0$ depends on the intermediate value theorem, which in turn depends on the completeness of $\mathbb{R}$. Over the rationals, there is no $x$ with $2^x=3$, because ${\log }_{2}3$ is irrational. The completeness axiom ensures that every positive real has a real logarithm, closing the gap that would otherwise prevent the inverse from being defined on all of $(0,\infty )$.

Historical & philosophical context [Master]

Napier 1614 Mirifici logarithmorum canonis descriptio ^[Napier1614] introduced logarithms as a computational device for reducing astronomical calculations. Napier's construction differed from the modern one: he defined logarithms via a kinematic model comparing a moving point on a line segment (producing a geometric progression) with a point moving along an infinite line (producing an arithmetic progression). Briggs 1617 Logarithmorum chilias prima adapted Napier's tables to base $10$, producing the common logarithms that dominated computation for three centuries. Euler 1748 Introductio in analysin infinitorum ^[Euler1748] identified logarithms as the inverse of exponentials, derived the laws of logarithms from the laws of exponents, and proved the change-of-base formula. The integral definition $\ln x={\int }_1^{x}dt/t$ appeared in the work of Gregory de Saint-Vincent 1647 Opus geometricum and was made rigorous by Mercator 1668 Logarithmotechnia. The connection between logarithms and prime numbers, culminating in the prime number theorem, was established by Hadamard 1896 Sur la distribution des zeros de la fonction $\zeta (s)$ et consequences arithmetiques (Bull. Soc. Math. France 24) and de la Vallee Poussin 1896 Recherches analytiques sur la theorie des nombres premiers (Ann. Soc. Sci. Bruxelles 20).

Bibliography [Master]

@book{Napier1614,
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  address = {Edinburgh},
  year = {1614}
}

@book{Briggs1617,
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  title = {Logarithmorum chilias prima},
  publisher = {privately printed},
  address = {London},
  year = {1617}
}

@book{Euler1748,
  author = {Euler, Leonhard},
  title = {Introductio in analysin infinitorum},
  publisher = {Marcus-Michaelis Bousquet},
  address = {Lausanne},
  year = {1748}
}

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  title = {A Course of Pure Mathematics},
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}

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}