Exponential & Logarithmic Equations

⚡ Key Takeaways

Exponential equations have the unknown in an exponent ( $a^x = b$ ); logarithmic equations have the unknown inside a logarithm ( $\log_a x = b$ ). The master technique: exponentials and logarithms are inverse operations, so applying one undoes the other. Used everywhere variables grow or decay multiplicatively — finance, biology, physics, chemistry. Formalized through the work of Napier, Briggs, and Euler (1614–1748).

💡 The Core Insight

Exponentiation and taking logarithms are mutually inverse. If a variable is trapped in an exponent, take a logarithm to free it; if a variable is trapped inside a logarithm, exponentiate to free it. Every solution technique on this page is a disciplined application of that single idea.

DefinitionExponential and Logarithmic Functions

For a fixed base $a$ with $a > 0$ and $a \neq 1$ :

1

\quad f(x) = a^{x}, \qquad x \in \mathbb{R}, \quad f(x) > 0

Derivation hint: extend integer powers $a^n = a\cdot a\cdots a$ to rational $a^{p/q} = \sqrt[q]{a^p}$ , then to reals by continuity. The logarithm base $a$ is its inverse:

2

\quad y = \log_a x \iff a^{y} = x, \qquad x > 0

Derivation hint: "the logarithm answers the question: to what power must I raise $a$ to obtain $x$ ?" Symbols: $a$ = base (positive, $\neq 1$ ); $x$ = argument/input; $y$ = exponent/output. The condition $a \neq 1$ is required because $1^y = 1$ for all $y$ , so $1^y = x$ has no unique solution.

DefinitionThe Two Equation Types

An exponential equation has the unknown in an exponent:

3

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

Derivation hint: any equation reducible to "base to a variable power equals a number." A logarithmic equation has the unknown inside a log:

4

\quad \log_a\big(h(x)\big) = c

Derivation hint: any equation reducible to "log of a variable expression equals a number."

TheoremFundamental Solving Identities

The following hold whenever both sides are defined ( $x,y > 0$ ):

5

\quad \log_a(xy) = \log_a x + \log_a y

Derivation hint: let $x=a^m,\,y=a^n$ ; then $xy=a^{m+n}$ , so $\log_a(xy)=m+n$ .

6

\quad \log_a\!\left(\tfrac{x}{y}\right) = \log_a x - \log_a y

Derivation hint: same substitution with division: $x/y = a^{m-n}$ .

7

\quad \log_a\!\left(x^{\,r}\right) = r\,\log_a x

Derivation hint: $x^r = (a^m)^r = a^{mr}$ , so the exponent is $mr = r\log_a x$ .

8

\quad \log_a x = \frac{\log_b x}{\log_b a} \qquad \text{(change of base)}

Derivation hint: set $y=\log_a x$ so $a^y=x$ ; take $\log_b$ of both sides and use [7].

9

\quad a^{\log_a x} = x \quad\text{and}\quad \log_a\!\left(a^{x}\right) = x

Derivation hint: direct statement that $a^x$ and $\log_a x$ are inverse functions.

ProofSolving the Generic Exponential Equation

a^{g(x)} = b

Step 1. Begin with $a^{g(x)} = b$ where $a,b>0$ and $a\neq 1$ . Both sides are positive, so taking a logarithm is valid.

Step 2. Apply $\log_a$ to both sides:

10

\quad \log_a\!\left(a^{g(x)}\right) = \log_a b

Derivation hint: logarithm is a one-to-one function, so applying it preserves equality.

Step 3. Use the inverse identity [9] on the left:

11

\quad g(x) = \log_a b

Derivation hint: $\log_a(a^{g(x)}) = g(x)$ exactly.

Step 4. If $a$ is not a convenient base, use change of base [8] to convert to natural or common logs:

12

\quad g(x) = \frac{\ln b}{\ln a}

Derivation hint: $\log_a b = \ln b / \ln a$ . Then solve the (now algebraic) equation $g(x) = \text{const}$ for $x$ . $\blacksquare$

∎

ProofSolving the Generic Logarithmic Equation

\log_a(h(x)) = c

Step 1. Begin with $\log_a(h(x)) = c$ . The domain requires $h(x) > 0$ — record this restriction now.

Step 2. Exponentiate both sides with base $a$ (apply $a^{(\cdot)}$ ):

13

\quad a^{\log_a(h(x))} = a^{c}

Derivation hint: $a^{(\cdot)}$ is one-to-one, so equality is preserved.

Step 3. Use the inverse identity [9] on the left:

14

\quad h(x) = a^{c}

Derivation hint: $a^{\log_a u} = u$ .

Step 4. Solve the algebraic equation $h(x) = a^c$ , then check every candidate against $h(x)>0$ ; discard extraneous roots. $\blacksquare$

∎

mathematician · Leonhard Euler | 1707–1783

Though John Napier invented logarithms (1614) as a computational aid, it was Leonhard Euler who in the 1730s–40s recast the subject as the modern theory of functions. He treated $a^x$ as a genuine function of a real variable, established the central role of the base $e \approx 2.71828$ , popularized the notation $f(x)$ , $e$ , and the use of $\log$ , and proved the foundational relations (including $e^{ix} = \cos x + i\sin x$ ). Euler's Introductio in analysin infinitorum (1748) is where exponentials and logarithms first appear as the inverse-function pair we teach today.

🧠 Intuition

Think of exponentiation as repeated multiplication and the logarithm as counting how many multiplications were needed. If bacteria double every hour, then after $t$ hours there are $2^t$ times as many — the exponential. To answer "how many hours until they multiply 1000-fold?" you must invert this: $\log_2 1000 \approx 9.97$ hours. The logarithm is a "multiplication-counter." A second picture: logarithms turn a multiplicative world into an additive one. On a logarithmic scale, multiplying by 10 means stepping the same distance every time — which is exactly why the Richter scale, decibels, and pH all compress huge multiplicative ranges into manageable additive numbers.

Figure

Exponential

y=2^x

and its inverse

y=\log_2 x

, mirrored across

y=x

ExampleExponential Equation — Same Base

Given: Solve $3^{\,2x-1} = 81$ .

Step 1. Express the right side as a power of the base: $81 = 3^4$ .

15

\quad 3^{\,2x-1} = 3^{4}

Derivation hint: $3^4 = 81$ by direct computation.

Step 2. Since the bases match and $a^x$ is one-to-one, equate exponents:

16

\quad 2x - 1 = 4

Derivation hint: $a^m = a^n \Rightarrow m = n$ for $a\neq 1$ .

Step 3. Solve: $2x = 5 \Rightarrow x = 2.5$ .

\boxed{x = \tfrac{5}{2}}

Check: $3^{2(2.5)-1} = 3^{4} = 81$ . ✓

ExampleLogarithmic Equation — With Extraneous Root

Given: Solve $\log_2(x) + \log_2(x-2) = 3$ .

Step 1. Note the domain: need $x>0$ and $x-2>0$ , so $x > 2$ .

Step 2. Combine using the product rule [5]:

17

\quad \log_2\big(x(x-2)\big) = 3

Derivation hint: $\log_a u + \log_a v = \log_a(uv)$ .

Step 3. Exponentiate (base 2) using [13]–[14]:

18

\quad x(x-2) = 2^{3} = 8

Derivation hint: $\log_2 N = 3 \Rightarrow N = 2^3$ .

Step 4. Solve the quadratic $x^2 - 2x - 8 = 0 \Rightarrow (x-4)(x+2)=0$ , giving $x = 4$ or $x = -2$ .

Step 5. Apply the domain $x>2$ : discard $x=-2$ (extraneous — it makes $\log_2(-2)$ undefined).

\boxed{x = 4}

Check: $\log_2 4 + \log_2 2 = 2 + 1 = 3$ . ✓

⚠️ The Three Deadly Errors

1. " $\log(x+y) = \log x + \log y$ " — FALSE. The product rule [5] applies to $\log(xy)$ , never to $\log(x+y)$ . There is no rule that simplifies the log of a sum. 2. Forgetting domain checks. Logs are only defined for positive arguments. Algebraic manipulation (squaring, combining logs) can create extraneous roots that violate $h(x) > 0$ . Always check. 3. " $\dfrac{\log x}{\log y} = \log x - \log y$ " — FALSE. The quotient rule [6] says $\log(x/y) = \log x - \log y$ . Dividing two logarithms is change of base [8], an entirely different operation.

CounterexampleWhy

\log(x+y) \neq \log x + \log y

Take base 10, $x = y = 1$ :

Left side: $\log_{10}(1+1) = \log_{10} 2 \approx 0.301$ .
Right side: $\log_{10} 1 + \log_{10} 1 = 0 + 0 = 0$ .

Since $0.301 \neq 0$ , the "rule" is decisively false. One numerical counterexample is enough to kill a proposed identity forever.

connection

Exponential and logarithmic equations sit at the crossroads of several core topics. They are inverse functions, so they directly extend Inverse Functions. Solving them often produces an Quadratic Equations in disguise (as in Example 2). The base $e$ and natural logarithm $\ln$ link this material to Calculus, where $\tfrac{d}{dx}e^x = e^x$ makes $e$ the natural base. The growth/decay applications connect to Geometric Sequences (discrete analog) and Differential Equations (continuous analog $\tfrac{dy}{dt}=ky$ ). Logarithmic scales tie directly into Scientific Notation.

🔬 Where this appears

⏳ History

- 1614 — John Napier publishes Mirifici Logarithmorum Canonis Descriptio, introducing logarithms to simplify astronomical multiplication.

- 1617 — Henry Briggs proposes base-10 ("common") logarithms after meeting Napier; publishes extensive tables.

- 1620 — Joost Bürgi independently publishes his own logarithm tables, developed before Napier but printed later.

- 1622 — William Oughtred invents the slide rule, a physical embodiment of logarithmic addition, in use for over 350 years.

- 1647 — Grégoire de Saint-Vincent and Alphonse de Sarasa discover that the area under the hyperbola

y=1/x

behaves like a logarithm (the natural log).

- 1683 — Jacob Bernoulli, studying compound interest, encounters the limit defining

e

.

- 1731 — The constant

e

first appears in a letter; Euler later names it.

- 1748 — Euler's Introductio in analysin infinitorum establishes

a^x

and

\log_a x

as inverse functions and centers the theory on

e

.

- 1972 — The HP-35 pocket calculator makes log tables and slide rules obsolete overnight.

open-problem · Beyond the Elementary Toolbox

Not every exponential equation can be solved with logarithms alone. Equations that mix a variable in both a polynomial and an exponent — such as $x e^{x} = c$ — cannot be solved using elementary functions. Their solution requires the Lambert $W$ function, defined as the inverse of $w \mapsto w e^{w}$ , which is not part of the standard high-school toolkit. Likewise, equations like $2^x = x^2$ have solutions that mix algebraic and transcendental behavior and must be found numerically or via $W$ . More deeply, questions about which exponential equations have closed-form solutions belong to transcendence theory — the Gelfond–Schneider theorem (1934) proves, for instance, that $2^{\sqrt 2}$ is transcendental, a result with no elementary derivation at all.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also