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Understanding exponents, roots, logarithms


    what you'll learn...

overview

This topic is little advanced for high school students. The various possibilities for roots and logarithms are discussed.

 •  root with negative power (4-2)

 •  root with power 0 (40)

 •  logarithm with fraction base (log128)

 •  logarithm of negative value (log2(-4))

 •  logarithm with negative base (log-2(-8))

 •  logarithm with 0 base (log02)

 •  logarithm with 1 base (log12)

Note: Some of the listed ones are not defined and has no meaning. Read through the lesson to understand the details.

first principles

revising the basics

Exponent is
(base)power=exp. result

If exponentiation result and power are given, then
base=resultpower
This inverse is called the "root".

If exponentiation-result and base are given, then
power=logbase(exp.result)
This inverse is called the "logarithm".

all combinations and their uses

Consider (base)power=exp. result

base and power can be one of positive integers, negative integers, 0, 1, or fractions.

Let us consider the important combinations one by one and understand some properties of the inverses "root" and "logarithm".

positive is simple enough

Consider (base)power=exp. result

Given that base and power are positive integers. The exp. result a positive integer".

negative base - not so simple

Given that, base is a negative integer and power is a positive integer. The exp. result can be a positive or negative value".

The negative value of the base is shown as -a with a positive integer a.

The exponentiation is
(-a)b=d
The result d can be a positive or a negative value.

The equation can be simplified to
(-a)b=(-1)bab=(-1)bc
and so, the value c is always positive.

In dealing with a negative base, the sign is separated out.

The following are noted.

 •  In the inverse "root", the exponentiation result (number c) is a positive number. Negative values for exponentiation result are studied separately in the higher level mathematics (complex numbers).

 •  In the inverse "logarithm", the base (number a) is a positive number. Negative bases are not studied, as the negative base is handled as power of -1 as shown.

 •  In the inverse "logarithm", the exponentiation result (number c) is a positive number. log of negative values are not defined.

Note 1: If the base is a positive value, then the exp. result is always positive. So logarithm of negative values is not defined.

Note 2: If we encounter log-ve a(+ve c)=b or log-ve a(-ve c)=b, then convert them into the equivalent exponent equation ab=c and solve for b.

negative power

Given that base is a positive integer and power is a negative integer. exp. result is "a positive fraction".

a-b=1ab.

So, the result is a positive fraction.

Given that base is a positive integer and power is a negative integer.

The negative value of power is shown as -b with a positive integer b.

The exponentiation is
a-b=1ab=(1a)b=c=1d
The result c is a positive fraction.

The following are noted.

 •  In the inverse "root", the power is always a positive number. Roots with negative powers are not studied.

Note 1: If root to the negative power is encountered, then convert that into its equivalent exponentiation equation ab=c and solve for the unknown value.

Note 2: In the inverse "logarithm", the base can be a fraction as given in the equation with 1a. This is explained along with the generic form of base pq in the next page.

fraction as base

Given that base is a positive fraction.
ab=(pq)b=c

To find the inverse "logarithm with base fraction pq --- "approach the log as the equivalent exponent". Logarithm with a fraction as base is not studied.

Consider (base)power=exp. result

Given that base is a fraction.
The exponentiation is
ab=(pq)b=c=pbqb

The result c is a fraction.
The following are noted.

 •  The inverse "root", the numerator and denominator can be independently considered as cb=numeratorbdenominatorb.

 •  The inverse "logarithm", logac, though the base a can be a fraction, it is not formally studied.

If we encounter logac where the base a is a fraction, then that is converted into an equivalent equation logac=logbc÷logba and solved.

fraction as power

Given that power is a positive fraction.

ab=apq=c

The inverse : "root" can be
cpq=a
cqp=a

Consider (base)power=exp. result

Given that power is a fraction.
The exponentiation is
ab=(a)pq=c=apq

The following are noted.

 •  In the inverse "root", cb, the power b=pq can be a fraction. The same is considered as cqp.

 •  In the inverse "logarithm", logac, the expression involves base a and value c and the variable under consideration is the result -- which can be a fraction.

base 0

Consider (base)power=exp. result

Given that base is 0 and power is a positive-non-zero value.
0b=0

The following are noted.

 •  0b=0.

 •  In ab=0b=c=0, the value of c can never be a non-zero value. So log to the base 0 is not defined for any non-zero values .

 •  In Further more, ab=0b=c=0, the value of b can be any number and does not solve to one value (meaning it has many solutions). So log00=b is also not defined.

power 0

Consider (base)power=exp. result

Given that base is a positive non-zero value and power is 0.
a0=1

The following are noted

 •  loga1=0

 •  In ab=a0=c=1, the value of c can never be other than 1. So c0 is not defined for any value other than 1.

 •  In ab=a0=c=1, the value of a can be any number and does not solve to one value (meaning it has many solutions). So, 10=a is also not defined as a can be any number.

base 1

Consider (base)power=exp. result

Given that base is 1
1b=1

The following are noted.

 •  1b=1. some additional learnings will be introduced in the higher mathematics, complex numbers

 •  In ab=1b=c=1, the value of c can never be any value other than 1. So log to the base 1 is not defined for any numbers that is not 1.

 •  In ab=1b=c=1, the value of b can be any number and does not solve to one value (meaning it has many solutions). So, log11=b is also not defined as b can be any number.

summary

Consider (base)power=exp. result OR ab=c

Summary of what we have learned in understanding roots and logarithms.

 •  log-ve ac=b is not formally studied, instead convert to exponent form ab=c to solve for b.

 •  c-ve b=a is not formally studied, instead convert to exponent form ab=c to solve for a.

 •  loga=p/qc=b is not formally studied, instead convert to exponent form ab=c to solve for b.

 •  log+ve a(-ve c)=b is not defined, as no value of b satisfies (+ve a)b=-ve c.

 •  c0=a is not defined, as no value of a satisfies a0=c for c1 and a takes many possible values if c=1

 •  log0c=b is not defined, as no value of b satisfies 0b=c for c0 and b takes many possible values if c=0.

 •  log1c=b is not defined, as no value of b satisfies 1b=c for c1 and b takes many possible values if c=1.

Outline

The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

    →   Representation of Exponents

    →   Inverse of exponent : root

    →   Inverse of exponent : Logarithm

    →   Common and Natural Logarithms

    →   Exponents Arithmetics

    →   Logarithm Arithmetics

    →   Formulas

    →   Numerical Expressions

    →   PEMA / BOMA

    →   Squares and Square roots

    →   Cubes and Cube roots