Math: Exponentials and Logarithms

This is Unit Twelve of ACT Preparation: exponentials and logarithms. Unit eleven studied curves built by adding powers; this unit studies growth that multiplies instead of adding. One idea runs through it: an exponential multiplies by a constant factor for each step, and a logarithm is the exponent that undoes it. By the end you will be able to write an exponential model from a percent change. To tell linear from exponential in a table. To use compound interest and the number e. To evaluate and rewrite logarithms. And to solve exponential and logarithmic equations, including half-life problems.

Nine entries on exponential functions: growth, decay, tables, interest and the graph.

A variable in the exponent.

ACT 1101 Exponential Functions

An exponential function has the form a times b to the x, with b positive and not one. The constant a is the initial value, the output at x equals zero: 500 times 1.04 to the x starts at 500 and is multiplied by 1.04 each step. The variable sits in the exponent; x squared is a different family.

Growing or decaying.

ACT 1102 Growth and Decay

When b is greater than one, the function grows; when b is between zero and one, it decays toward zero. Two hundred times one half to the x goes 200, 100, fifty, twenty-five. The base alone decides, and decay never reaches zero.

Factor and rate.

ACT 1103 Growth Factor and Growth Rate

The base b is the growth factor, and the growth rate is b minus one: five percent growth is a factor of 1.05, and five percent decay a factor of 0.95. Use the factor, not the rate: multiplying by 0.08 would shrink sales to eight percent of their size.

A model from a percent change, step by step.

ACT 1104 Writing a Model From a Percent Change

Find the initial value, convert the percent change to a factor, one plus or one minus the rate, and write a times b to the t, with t counted in the rate's unit. A town of 12,000 growing three percent a year is 12,000 times 1.03 to the t, about 12,731 after two years. A $25,000 car losing fifteen percent a year is worth $18,062.50 after two. Each year's percent applies to the new amount, so two years of three percent is a factor of 1.0609: unit seven's rule for successive changes, written as a function.

Linear or exponential.

ACT 1105 Linear or Exponential

Over equal steps in x, a linear function adds a constant difference, while an exponential function multiplies by a constant ratio. Three, seven, eleven, fifteen adds four: linear. Three, six, twelve, twenty-four doubles: exponential. In this course the rule rests on the anchor alone, with no witness stating it as a table test.

Classifying a table, step by step.

ACT 1106 Telling Linear From Exponential in a Table

Confirm the inputs are equally spaced, subtract consecutive outputs, and if the differences are unequal, divide them. Eighty, forty, twenty, ten has unequal differences but ratios of one half throughout, so it is exponential, eighty times one half to the x. Five, eight, eleven, fourteen has differences of three, so it is linear, 3x plus five. This rests on the anchor alone, as entry 1105 says, and a table can look linear for two steps and diverge later, so check every pair.

Compound interest.

ACT 1107 Compound Interest

Compound interest grows a principal P at annual rate r, compounded n times a year for t years: P times one plus r over n, to the n t. One thousand dollars at six percent compounded monthly for two years is 1,000 times 1.005 to the twenty-fourth, about $1,127.16. The rate per period is r over n.

The number e.

ACT 1108 The Number e

The number e, about 2.718, is the base of continuous growth, and continuously compounded interest is P times e to the r t. One thousand dollars at six percent for two years becomes about $1,127.50. E is a constant like pi, not a variable.

The exponential graph.

ACT 1109 The Exponential Graph

With a positive, the graph of a times b to the x passes through zero, a, stays above the x-axis, and approaches the asymptote y equals zero. Two to the x passes through zero, one and one, two. A vertical shift moves the asymptote: two to the x minus three approaches negative three.

Seven entries on logarithms: what they are, their rules, and their graph.

A logarithm is an exponent.

ACT 1110 What a Logarithm Is

The logarithm base b of x is the exponent b must be raised to in order to give x. Log base two of eight is three, because two cubed is eight; log base five of one is zero. The input must be positive, since b to any power is never zero or negative.

The two common bases.

ACT 1111 Common and Natural Logarithms

The common logarithm has base ten, and the natural logarithm, written l n, has base e. Log of 1,000 is three, log of 0.01 is negative two, and l n of e to the fifth is five. A log written with no base means base ten.

Evaluating logarithms, step by step.

ACT 1112 Evaluating Logarithms

Set the log equal to y, rewrite in exponential form, write the input as a power of the base, and read y. Log base three of eighty-one is four; log base two of one sixteenth is negative four; log base four of two is one half; and log base nine of twenty-seven is three halves, using the common base three. If there is no common base, the change-of-base formula gives a decimal.

The rules of logarithms.

ACT 1113 Product, Quotient and Power Rules

The log of a product is the sum of the logs, of a quotient the difference, and of a power the exponent times the log. Log base two of thirty-two is log base two of four plus log base two of eight, two plus three, five. The log of a sum has no rule.

Changing the base.

ACT 1114 Change of Base

Any logarithm can be rewritten in another base: log base b of x equals log x divided by log b. Log base two of ten is one divided by 0.3010, about 3.32. Divide the logs; the log of the quotient is a different number.

Expanding and condensing, step by step.

ACT 1115 Expanding and Condensing Logarithms

To expand, split quotients into differences, products into sums, and bring exponents down, writing roots as fractional exponents first; to condense, run the steps in reverse. Log of x cubed root y over z squared expands to 3 log x plus one half log y minus 2 log z. The power rule applies only to a power of the whole input: log of 3x squared is log three plus 2 log x.

The logarithm graph.

ACT 1116 The Logarithm Graph

The graph of log base b of x reflects the exponential over the line y equals x. It passes through one, zero, has the vertical asymptote x equals zero, and its domain is positive x. Log base two passes through two, one and eight, three. Domain and range swap with the exponential's.

Six entries on solving exponential and logarithmic equations, and on half-life.

Solving with a common base.

ACT 1117 Solving by a Common Base

If both sides can be written as powers of the same base, set the exponents equal. Four to the x plus one equals eight to the x becomes two to the 2x plus two equals two to the 3x, so x is two. Rewrite both sides fully before comparing.

Solving with logarithms.

ACT 1118 Solving With Logarithms

When no common base exists, isolate the exponential, take the log of both sides, and bring the exponent down. Three to the x equals twenty gives x equals log twenty over log three, about 2.73. Isolate first: in five times three to the x equals one hundred, divide by five before taking logs.

An exponential equation, step by step.

ACT 1119 Solving an Exponential Equation

Isolate the exponential, look for a common base, otherwise take logs, then solve and check. Two times five to the x minus one, plus three, equals fifty-three isolates to five to the x minus one equals twenty-five, so x is three. Six to the 2x equals fifteen has no common base, so x is log fifteen over 2 log six, about 0.76. Do not take logs before isolating: the log of a sum does not split.

A logarithmic equation, step by step.

ACT 1120 Solving a Logarithmic Equation

Combine the logs on each side into one, rewrite in exponential form or set inputs equal, solve, and check that every log's input in the original is positive. Log base two of x plus log base two of x minus two equals three becomes x times x minus two equals eight, so four or negative two. Negative two makes log base two of x undefined, so only four solves it. Combining the logs hides the original domain, which is why the check is required.

Half-life and doubling time.

ACT 1121 Half-Life and Doubling Time

A quantity with half-life h halves every h units of time; one with doubling time d doubles every d. Eighty grams with a five-day half-life leaves ten grams after fifteen days. Count half-lives first: fifteen days is three half-lives, not fifteen.

A half-life problem, step by step.

ACT 1122 A Half-Life Problem

Write the half-life model with time and half-life in the same unit, substitute, and either evaluate or divide and take logs. In an invented isotope with an eight-hour half-life, 400 milligrams decays to fifty, one eighth, after three half-lives, twenty-four hours. After twelve hours, one and a half half-lives, about 141 milligrams remain, not the 150 a straight line would suggest.

Look at what you can now do. You can write an exponential model from a percent change, using the factor and not the rate. You can tell linear from exponential by differences and ratios. You can use compound and continuous interest, and read the exponential graph and its asymptote. You can evaluate logarithms as exponents, and expand, condense and change their base. And you can solve exponential and logarithmic equations, isolating first and checking every input. Each time, the factor multiplied and the logarithm undid it. Unit thirteen turns to geometry: angles, triangles and shapes.