Exponential and Logarithmic Functions

This is Unit Eight of College Algebra: exponential and logarithmic functions. The capstone gives the unit its key in one sentence: every logarithm law is an exponent law seen in the mirror. Carry that from the start and the second half of this unit stops being a list to memorise. The first family multiplies rather than adds. Each step forward multiplies the output by a fixed amount, and that is a different kind of change from everything in the course so far — different enough that over any long run it overtakes linear change regardless of the numbers involved. The second family answers the question the first one raises: what exponent produces this result. That is all a logarithm is, and the entries say so repeatedly, because the notation obscures a question most people can already answer. Three named errors sit together in one topic, and all three come from expecting logarithms to distribute the way multiplication does. They convert products into sums — which is not distribution, and does not extend to sums. By the end you will be able to do five things. One: say what separates exponential change from linear, and why the ratio wins. Two: move a statement between the two forms in both directions. Three: state the three laws and the errors each invites. Four: solve an exponential equation whose bases will not match. Five: say why logarithmic answers get audited.

Four entries introducing a genuinely new kind of change. Everything in the course until now added a fixed amount per step. This family multiplies by a fixed amount per step, and the third entry makes the consequence explicit.

The family.

Alg 411 Exponential Functions

A fixed base raised to the input, with a coefficient in front. Every unit step forward multiplies the output by the base.

Which direction it goes.

Alg 412 Growth and Decay

A base greater than one makes the outputs grow; a base between zero and one makes them shrink. One number decides the whole character.

How it differs from a line.

Alg 414 Linear versus Exponential Change

A line adds the same amount each step; this family multiplies by the same amount each step. Over a long enough run the multiplying always overtakes the adding, whatever the two rates are.

Computing one.

Alg 416 Evaluating Exponential Functions

Put the input into the exponent and apply the rules from Unit One. The exponent attaches to the base — the coefficient in front is never raised.

The application that produced this family historically, and then something unexpected: push the compounding to its limit and a specific number falls out. That number turns out to matter far beyond finance.

The application.

Alg 418 Compound Interest

A starting amount grown at a stated annual rate, compounded a set number of times per year. Two conversions come first — the percentage into a decimal, and the years into periods — and both are done before anything else.

The number that falls out.

Alg 420 The Number e

Compound more and more often and the growth factor settles toward a specific value, roughly two point seven one eight. It never terminates and never repeats — an irrational number arriving out of a finance calculation.

Compounding without pause.

Alg 421 Continuous Growth and Decay

Using that number as the base gives growth compounded continuously — the limit of compounding more and more frequently, written as a single formula.

One entry, and the feature worth holding is the horizontal asymptote — the graph approaches zero forever on its shrinking side and never arrives, which is what keeps the outputs positive throughout.

The picture.

Alg 423 Graphs of Exponential Functions

Every graph in the family passes through the same point on the vertical axis, accepts any input, and produces only positive outputs — running along the horizontal axis without ever touching it.

The second family, and it is the first one reversed. A logarithm asks what exponent produces a given result — that is the entire idea, and the notation is the only difficult part of it.

The question it asks.

Alg 426 The Logarithm

Given a base and a result, the logarithm is the exponent producing it. Two statements — one written as a logarithm, one as a power — say exactly the same thing.

The relationship.

Alg 427 Logarithms and Exponentials Are Inverses

Each logarithm reverses its exponential, so their graphs reflect across the diagonal and each one's inputs are the other's outputs. This is Unit Five's inverse relationship, on a specific pair.

Two special cases.

Alg 429 Common and Natural Logarithms

Base ten and the base from the last topic each get their own shorthand. A logarithm written with no base shown means base ten in this course.

Three entries on handling both forms fluently. Converting between them, evaluating without a calculator by asking the question aloud, and the graph — which is the previous family's picture reflected.

Converting.

Alg 431 Converting Between Forms

The two forms are one statement in two costumes. The base stays where it is in both, and the exponent and result exchange positions around it — which makes the conversion mechanical once you see what stays fixed.

Evaluating one.

Alg 433 Evaluating Logarithms Mentally

Ask what power of the base produces the input, out loud. Most values people find opaque in the notation are immediate once stated as that question.

The picture.

Alg 435 Graphs of Logarithmic Functions

Only positive inputs allowed, passing through the point where the output is zero, and falling along the vertical axis — the exponential graph reflected across the diagonal, exactly as the inverse relationship requires.

Six entries, alternating between a law and a misreading of it. Each law is an exponent rule reflected: multiplication adds exponents, so a logarithm turns a product into a sum. The errors all come from expecting distribution instead.

The first law.

Alg 446 The Product Rule for Logarithms

The logarithm of a product equals the sum of the logarithms. And the reason is Unit One's product rule — multiplying powers adds exponents, and a logarithm is an exponent.

The first error.

Alg 447 The Split-Sum Log Error

Splitting the logarithm of a sum. The law converts multiplication into addition; it does not convert addition into anything, and a sum inside stays where it is.

The second error.

Alg 448 The Multiplied-Logs Error

Turning the logarithm of a product into a product of logarithms. The law produces a sum — the operation changes, which is the whole point of it.

The second law.

Alg 450 The Quotient Rule for Logarithms

The logarithm of a quotient equals the difference of the logarithms — the reflection of the exponent rule where dividing subtracts.

The third error.

Alg 451 The Divided-Logs Error

Reading a quotient of two logarithms as the logarithm of a quotient. Those are different expressions, and the second topic from here explains what the first one actually is.

The third law.

Alg 453 The Power Rule for Logarithms

A power inside the logarithm becomes a multiplier outside it. The exponent has to cover the entire input — one raised to a power inside a larger expression does not qualify.

Running the laws in both directions, and then a formula that looks precisely like the error two entries ago — and is not, which is why the entry names the distinction directly.

Both directions.

Alg 455 Expanding and Condensing Logarithms

Break one logarithm of a complicated expression into simple pieces, or run the same laws backwards to collect pieces into one. Only products, quotients and powers come apart — a sum inside stays whole.

The look-alike.

Alg 457 The Change-of-Base Formula

Any logarithm can be computed as a quotient of two logarithms in a borrowed base — which is what a quotient of logarithms actually is. It is a change of base, and never the logarithm of a quotient.

Two entries. When both sides can be written with the same base, the exponents can be equated directly. When they cannot, a logarithm applied to both sides brings the exponent down where it can be reached.

When the bases match.

Alg 463 The One-to-One Property of Exponential Functions

With one base on both sides, the exponents must be equal, and the equation reduces to something already solvable.

When they will not.

Alg 465 Solving Exponential Equations with Logarithms

Take a logarithm of each side, use the power law to bring the exponent down, and solve what remains. The logarithm applies to the entire side — applied to one term of a sum it produces a different equation.

Two entries, and the second contains the unit's checking requirement. This family refuses inputs that are not positive, and solving can produce candidates the original equation cannot accept.

The matching property.

Alg 467 The One-to-One Property of Logarithms

With one base on both sides, equal logarithms force equal inputs — the mirror of the property from the last topic.

Solving, and auditing.

Alg 468 Solving Logarithmic Equations

Condense to a single logarithm, convert to the other form, solve, then test every candidate. The check runs against the original equation, because that is where the restriction on the input lives.

One application, and its result is more surprising than the arithmetic — the time taken to double does not depend at all on how much you started with.

The span that does not depend on the start.

Alg 473 Doubling Time

Solving for when continuous growth reaches twice its starting value gives a span fixed by the rate alone. Match the logarithm to the base — the natural one for that base — and the exponent comes down in a single move.

The unit closes on the sentence the opening borrowed, now with all six laws behind it — and it explains why the second half never needed to be memorised as a separate list.

The capstone. One relationship.

Alg 480 The Inverse Kingdom

The course's fastest-growing functions paired with the functions that reverse them, and every logarithm law is an exponent law seen in the mirror. Every claim here carries an earlier number.

Five things you can now do. You can say what separates the two kinds of change — a constant difference per step against a constant ratio per step — and why the ratio overtakes the difference over any long enough run. You can move a statement between the two forms, knowing the base stays fixed while the exponent and result trade places around it. You can state the three laws and the errors each invites — products become sums, quotients become differences, powers come out front, and none of it is distribution. You can solve an exponential equation whose bases will not match, by taking a logarithm of each entire side and bringing the exponent down. And you can say why logarithmic answers get audited: this family refuses nonpositive inputs, and the restriction lives in the original equation.