Exponentiation answers the question: given a base and an exponent, what is the result? Logarithms reverse this process, answering: given a base and a result, what exponent produced it? This inverse relationship makes logarithms essential wherever exponential growth or decay appears — compound interest, radioactive decay, sound intensity, earthquake magnitude, and countless other phenomena.
The expression loga(b)=c states that ac=b. The logarithm extracts the exponent. Once this connection is clear, logarithms transform from mysterious notation into a natural extension of the exponential ideas already familiar from earlier algebra.
What is a Logarithm?
A logarithm answers the question: to what power must a base be raised to produce a given number? The expression loga(b) asks for the exponent that satisfies a?=b.
The notation loga(b)=c is read as "log base a of b equals c." This statement is equivalent to the exponential form ac=b. The two expressions encode identical information — one solved for the exponent, the other presenting the complete power relationship.
Consider log2(8)=3. This asks: what power of 2 gives 8? Since 23=8, the answer is 3. Similarly, log10(1000)=3 because 103=1000, and log5(25)=2 because 52=25.
The base sits as a subscript, the argument sits inside parentheses, and the output is the exponent. Every logarithm can be rewritten as an exponential equation, and every exponential equation can be rewritten using logarithms. This duality is the foundation of all logarithmic work — converting between forms is often the first step in solving problems.
Restrictions on Base and Argument
Not every combination of base and argument produces a valid logarithm. Two restrictions apply.
The base a must satisfy a>0 and a=1. A negative base would produce complex or undefined values for most exponents — (−2)1/2 is not a real number. A base of zero fails because 0c=0 for positive c, meaning no exponent could produce any result other than zero. A base of one fails because 1c=1 for all c, meaning every number would need the same logarithm, which is incoherent. Bases between zero and one are valid and produce decreasing logarithmic functions.
The argument b must satisfy b>0. No real exponent applied to a positive base can produce zero or a negative number. Since ac is always positive when a>0, logarithms of zero or negative numbers do not exist in the real number system. Attempting to compute log2(−4) or log3(0) has no solution.
These restrictions are not arbitrary conventions — they emerge directly from the behavior of exponential functions. The properties of logarithms and the shape of their graphs all follow from these fundamental constraints.
Key Logarithmic Values
Two values hold for every valid base and appear constantly in calculations.
The first: loga(1)=0 for any base a. This follows immediately from the exponential form. Since a0=1 for every positive a=1, the exponent that produces 1 is always zero. The point (1,0) lies on every logarithmic graph.
The second: loga(a)=1 for any base a. Since a1=a, raising the base to the first power returns the base itself. The exponent that produces the base is always one. The point (a,1) lies on every logarithmic graph.
These values serve as anchors. When sketching graphs, they provide two guaranteed points. When checking calculations, they offer quick verification — if a formula gives log3(1)=0, something has gone wrong.
Additional values follow patterns. For base 10: log10(10)=1, log10(100)=2, log10(1000)=3, and so on — each multiplication by 10 increases the logarithm by 1. For base 2: log2(2)=1, log2(4)=2, log2(8)=3, log2(16)=4. Recognizing powers of the base makes mental calculation possible.
Inverse Identities
Logarithms and exponentials undo each other. Two identities express this relationship and arise whenever the operations are composed.
The first identity: loga(ax)=x. Taking the logarithm of an exponential with the same base extracts the exponent. The expression log2(25)=5 without calculation — the logarithm simply returns what the exponent was. This works because logarithm asks "what power of a gives ax?" and the answer is obviously x.
The second identity: aloga(x)=x. Raising the base to a logarithmic power returns the original argument. The expression 3log3(7)=7 without calculation — the exponential returns what the argument was. This works because loga(x) is precisely the power that produces x when applied to base a.
These identities are not tricks to memorize but logical necessities. If f(x)=ax and g(x)=loga(x), then f and g are inverse functions: f(g(x))=x and g(f(x))=x. Each operation reverses the other.
In practice, these identities simplify expressions and solve equations. Recognizing when an expression contains a logarithm and exponential with matching bases allows immediate simplification.
Common and Natural Logarithms
Two bases appear so frequently that they receive special notation.
The common logarithm uses base 10 and is written log(x) without a subscript, or sometimes log10(x) for clarity. Base 10 aligns with the decimal system, making common logarithms natural for expressing orders of magnitude, pH calculations, and decibel scales.
The natural logarithm uses base e≈2.71828 and is written ln(x). The number e emerges from calculus — it is the unique base for which the derivative of ax equals ax itself. Natural logarithms dominate theoretical mathematics, physics, and any context involving continuous growth or decay.
Both notations are shorthand. The statement log(100)=2 means log10(100)=2. The statement ln(e3)=3 means loge(e3)=3. Converting between bases uses a formula covered in logarithm rules.
Properties of Logarithms
Logarithmic functions possess structural characteristics that determine their behavior across the entire domain.
The domain of loga(x) is all positive real numbers — only positive arguments are permitted. The range is all real numbers — logarithms can produce any output, positive or negative. As x approaches zero from the right, the logarithm decreases without bound. As x increases without bound, the logarithm also increases, but slowly.
Monotonicity depends on the base. When a>1, the function is strictly increasing: larger inputs produce larger outputs. When 0<a<1, the function is strictly decreasing: larger inputs produce smaller outputs. This property is critical when solving inequalities.
The one-to-one property follows from monotonicity: if loga(x)=loga(y), then x=y. No two distinct inputs can produce the same output. This property justifies a key technique in solving equations — when both sides of an equation are logarithms with the same base, the arguments must be equal.
Logarithm Rules
The rules of logarithms transform how arguments combine under the logarithm function. Each rule corresponds to an exponent law, inverted.
The product rule: loga(xy)=loga(x)+loga(y). Logarithms convert multiplication into addition. This follows from am⋅an=am+n — if the exponents add when bases multiply, then logarithms of products split into sums.
The quotient rule: loga(x/y)=loga(x)−loga(y). Division becomes subtraction. The power rule: loga(xn)=n⋅loga(x). Exponents come down as coefficients.
These rules allow expanding complex logarithms into simpler pieces and condensing sums of logarithms into single expressions. Both skills appear constantly when solving equations. The change of base formula converts logarithms from one base to another, enabling calculator computation for any base.
Logarithmic Equations
Equations involving logarithms appear in two main forms: those where the logarithm equals a constant, and those where logarithms appear on both sides.
An equation like log2(x)=5 converts directly to exponential form: x=25=32. The logarithm definition provides the solution method. An equation like log3(x−1)=log3(7) uses the one-to-one property: since the logarithms are equal and have the same base, the arguments must match, giving x−1=7 and x=8.
More complex equations require the rules to combine or separate logarithms before these techniques apply. Every solution must be checked against domain restrictions — the argument of every logarithm must be positive. Extraneous solutions that violate these restrictions must be rejected.
Logarithms also solve exponential equations when matching bases fails. An equation like 3x=7 has no integer solution, but taking logarithms of both sides and applying the power rule yields x=log(7)/log(3).
Logarithmic Inequalities
Solving logarithmic inequalities follows the same algebraic steps as equations, with one critical addition: the base determines whether the inequality direction is preserved or reversed.
When the base satisfies a>1, the logarithm is increasing. Larger arguments produce larger outputs, so inequality direction is preserved. If log2(x)>3, then x>23=8.
When the base satisfies 0<a<1, the logarithm is decreasing. Larger arguments produce smaller outputs, so inequality direction reverses. If log1/2(x)>3, then x<(1/2)3=1/8.
Domain restrictions apply throughout. Every argument must remain positive, and this constraint intersects with the algebraic solution. The properties page covers monotonicity in detail; the graphs page shows why direction reversal occurs visually.
Graphing Logarithmic Functions
The graph of y=loga(x) has a distinctive shape determined by the properties of the function.
Every logarithmic graph passes through (1,0) because loga(1)=0 for all bases. Every graph also passes through (a,1) because loga(a)=1. A vertical asymptote exists at x=0 — the graph approaches the y-axis but never touches it, plunging toward −∞ as x approaches zero from the right.
The base controls the overall direction. For a>1, the graph rises from left to right. For 0<a<1, the graph falls. Larger bases (when greater than one) produce flatter curves; the function grows more slowly.
Transformations shift, stretch, and reflect the basic shape. Horizontal shifts move the asymptote. Vertical shifts raise or lower the curve. Reflections flip across axes. Writing equations from transformed graphs requires identifying the new asymptote location and key points.