Relationship between Indices and Logarithmic Functions

This tutorial discusses about logarithms and the relationship between indices and logarithms.

Logarithms can be taken simply as the inverse of indices.

The meaning of Logarithm

If a^x = y such that a > 0, a ≠ 1 then log_a y = x

a^x = y ↔ loga y = x

Exponential Form

y = a^x

Logarithmic Form

log^a y = x

Note: The logarithm is the exponent.

Example:

Convert the following exponential (indices) form to the logarithmic form:

a) 4^{2 = 16}

b) 2⁵ = 32

c) 2^-1=¹₂

Answer:

a) 4² = 16
2 = log₄ 16 (the log is the indices or exponent)

b) 2⁵ = 32
5 = log₂ 32

2^-1=¹₂

-1=log₂¹₂

Example:

Convert the logarithmic form below to (indices) or exponential form

a) 3 = log₂ 8

b) 2 = log₅ 25

Solution:

a) 3 = log₂ 8
2³ = 8

b) 2 = log₅ 25
5² = 25

Observe the following:

• Since a¹ = a, loga a = 1
• Since a⁰ = 1, log_a 1 = 0
• Loga 0 is undefined
• Logarithms of all negative numbers are undefined.
• The base of logarithms can be any positive number apart from 1.
• Logarithms to the base 10 are referred to as common logarithms and are represented by log₁₀ or log.
• Logarithms to the base e are referred to as natural logarithms and are denoted by log_e or ln.

Function Notation with Logs and Exponentials

Function notation is utilized constantly in science to express functions that are made up of logs and exponents. We learn to make use of function notation with logs and exponentials to be able to tackle problems like calculating compound interest. We can solve these problems written in function notation with logs and exponentials making use of procedures from the calculation of exponential and log equations.

Before this topic, it is presumed that you are already aware of how to calculate law of indices for positive integer powers and

• How to work with the arithmetic of integers and fractions.
• How to work with basic algebra.
• And you are already conversant with rounding numbers correct to a specific number of decimal places.

Drive

Indices make available a compact algebraic notation for repeated multiplication. For instance, is it much easier to write 35 than 3 × 3 × 3 × 3 × 3.

Immediately the index notation is introduced, the index laws arise naturally when simplifying numerical and algebraic expressions. Therefore, the simplification 25 × 23 = 28 quickly leads to the rule am × an = am + n, for every positive integers m and n.

As frequently done in mathematics, it is natural to ask questions like:

• Can we offer meaning to the zero index?
• Can we offer meaning to a negative index?
• Can we offer meaning to a rational or fractional index?

These questions will be looked at here. In a lot of applications of mathematics, we can express numbers as powers of a few given base. We can alternate this question and ask, for instance, ‘What power of 2 produces 16? Our attention is then diverted to the index itself. This results to the notion of a logarithm, which is merely another name for an index.

Logarithms are made use of in a lot of places:

• decibels, that are utilized to measure sound pressure, are defined with logarithms
• the Richter scale, that is made use of to measure earthquake intensity, is defined with logarithms
• the pH value in chemistry, that is made use of in defining the level of acidity of a substance, is as well defined with the use of the notion of a logarithm.

When two measured quantities seem to be related by an exponential function, the parameters of the function can be calculated with the use of log plots. This is a very essential tool in experimental science.

Logarithms can be made use of in the calculation of equations like 2x = 3, for x.

In higher mathematics, proficiency in manipulating indices is indispensable, because they are used broadly in both differential and integral calculus. Therefore, to differentiate or integrate a function like , it is first essential to convert it to index form.

The function in calculus that is a multiple of its own derivative is an exponential or indices function. Functions like that are made use of in the modeling of growth rates in biology, ecology and economics, in addition to radioactive disintegration in nuclear physics.

Indices

You can remember that a power is the product of a particular number of factors, all of which are equivalent. For instance, 37 is a power, in which the number 3 is known as the base and the number 7 is known as the index or exponent.

In the tutorial, Multiples, Factors and Powers, the index laws below were established for positive integer exponents. Therefore with positive integers and rational numbers we obtain:

Index Laws

1. To multiply powers with equivalent base, add the indices. aman = am+n.

2. To divide powers with equivalent base, subtract the indices. ^{a^m}⁄_aⁿ= am − n, (provided m > n.)

3. To raise a power to a power, multiply the indices. (am)n = amn.

4. A power of a product is the product of the powers. (ab)m = ambm.

5. A power of a quotient is the quotient of the powers. (^a⁄_b)m=^{a^m}⁄_b^m , (provided b ≠ 0.)

These laws as well hold when a and b are real

For all intergers m and n and non-zero numbers a and b the following rules are applicable:

For all intergers m and n and non-zero numbers a and b the following rules are applicable:
Zero exponent		a0 = 1
Negative exponent		a−n =¹⁄_aⁿ
Index law 1	Product of power	aman = am+n
Index law 2	Quotient of power	am ÷ an = ^{a^m}⁄_aⁿ= am − n
Index law 3	Power of a power	(am)n = amn
Index law 5	Power of a quotient	(^a⁄_b) n = ^aⁿ⁄_bⁿ

Fractional Indices or exponent

We currently broaden our study of indices to involve rational or fractional indices. Particularly, can we give meaning to 4^¹⁄₂ ?

Now again, we would make use of the established index laws. Therefore, , squaring the expression we would obtain:

(4^¹⁄₂) 2 = 4^¹⁄₂ × 2 = 41 = 4.

Therefore, we define 4 ^¹⁄₂ to be √ 4 = 2.

Generally, we define a^¹⁄₂ = √ a for any positive number a.

Observe that we have defined a ^¹⁄₂ to be the positive square root. We perform this operation in order that there is just one value for a^¹⁄₂ .

Applying an equivalent argument, for consistency with the index laws, we define a ^¹⁄₃ = √ 3 , a ^¹⁄₄ = √ 4 , and so on.

Logarithms

It is simple to obtain values of x, like that 2x = 2 or 2x = 4, or 2x = 32. On the contrary, how do we compute the equation 2x = 10?

Problems like this come up naturally when we are computing exponential growth and decay.
In the example above, we give the formula for the mass of a radioactive substance to be M = 100 × (¹⁄₂) t g.

If we are asked the question, when is the mass equal to 30g, then we are required compute (¹⁄₂)t = 0.3 to obtain the time.

In the same way as taking a square root is the inverse process to squaring, taking a logarithm is the inverse process of taking a power.

Since 2³ = 8, we say that log₂ 8 = 3. This means that the logarithm is the index in the equation: 2³ = 8. This can be read as ‘the log of 8 to the base 2 is 3.’

To calculate the logarithm of a number a to the base b, we ask the question ‘what power do I raise b to, in order to get a?

Therefore, to obtain for instance, log3 243, we make use of the fact that 243 = 35, so log3 243 = 5.

The relationship linking logarithms and powers is:

x = loga y means y = ax.

The number is known as the base and ought to be a positive number. Again, since ax is positive, we can only obtain the logarithm of a positive number. We will presume from now on that both are positive, but can be negative.

Examples

a log₂ 32 = x	b log₈ = x	c log₂ x = 5
d log_x 16 = 2	e log₃₆ x = −	f log₇ x = 2

Note: The identities below exemplify the inverse operations of taking a power and taking a logarithm. These require to be adequately understood by students.

For x > 0,

2log₂ x = x.

In a more general form, for a > 0, x > 0,
alog_a x = x.

In the other direction, for any x,
log_a 2x = x.

In a more general form, for a > 0,
log_a ax = x.

It is significant for students to understand these two general identities very well.

Logarithms to the base 10

You will observe that in all the examples above, the values of the logarithms were rational numbers, which were not too hard to find. Assuming we wanted to know the value of log10 7? Therefore, we seek a number x such that 7 = 10x.

We can observe from the graph of y = 10x that a number like that lies between 0 and 1.

The calculator is capable to offer an approximate value of this number. It is shown in the module, The Real Numbers that numbers like this are irrational.

Therefore, to 4 decimal places, the calculator gives that log10 7 ≈ 0.8451.

The Logarithm Laws

Suppose a > 0 for the rest of this section.

Law 1
loga = 0 and loga a = 1
since a0 = 1, we have loga 1 = 0.
In the same way, since a1 = a, we have loga a = 1.

Law 2

If x and y are positive numbers, then loga xy = loga x + loga y This means, the logarithm of a product is the sum of the logarithms.

Assume x = ac and y = ad so that loga x = c and loga y = d.

Then xy = ac × ad =ac+d (by Index law 1) Therefore loga xy = loga ac+d = c + d = loga x + loga y

Law 3

If x and y are positive integers, then loga ^x⁄_y= loga x − loga y. That is, the logarithm of a quotient is the variation of their logarithms.

Assume x = ac and y = ad so that loga x = c and loga y = d.

Then ^x⁄_y =^{a^d}⁄_a^d
= ac−d (by Index law 2)
Therefore, loga = loga ac−d
= c − d
= loga x + loga y

Law 4

If x is a positive number, then loga^¹⁄_x = −loga x.

This follows from logarithm law 3 and logarithm law 1.

loga ^¹⁄_x = loga 1 − loga x (logarithm law 3)
= 0 − loga x (logarithm law 1)
= −loga x, as required.

Law 5

If x is a positive integer and n is any rational number, then loga (xn) = nloga x.

This entails from logarithm law 3 and logarithm law 1.

loga (xn) = loga ((ac)n)
= loga (acn) (by Index law 3)
= nloga x, as required.

EXAMPLE

Given log7 2 = α, log7 3 = β and log7 5 = , express in termd of α, β and :

a log7 6 b log7 6 c log7 ^¹⁵⁄₂

solution

a log7 6 = log7 (2 × 3)
= log7 2 + log7 3
= α + β

b log7 6 = log7 (3 × 25)
= log7 3 + log7 52
= β + 2

c log7 ^¹⁵⁄₂ = log7 15 − log7 2 = log7 (3 × 5) − log7 2 = log7 3 + log7 5 − log7 2 = β + − α

Practice Questions

Simplify:

a logb x2 + logb x3 − logb x4

b logk^a⁄_b + logk^b⁄_a

c logb (x2 − a2) − logb (x − a), if x > a

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Relationship between Indices and Logarithmic Functions