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Lesson 5, Topic 1
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# Relationship between Indices & Logarithms

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Recall the laws of indices;

1. $$\scriptsize a^x \: \times \: a^y = a^{x \:+\:y}$$

2. $$\scriptsize a^x \: \div \: a^y = a^{x \:-\:y}$$

3. $$\scriptsize a^0 = 1$$

4. $$\scriptsize \left (a^x \right)^y = a^{xy}$$

5. $$\scriptsize a^{-x} = \normalsize \frac {1}{a^x}$$

6. $$\scriptsize a ^{\frac{x}{y}} = \left ( \scriptsize \sqrt [y] {a} \right) ^x$$

7. $$\scriptsize a ^{\frac{1}{x}} = \sqrt [x] {a}$$

These laws are true for all values of x, y and a ≠0. It is common knowledge that scientific calculators have taken the place of logarithm and anti-logarithm tables. However, it is important to note that the theory of Logarithms is very relevant in science and technology.

Note that in index notation 34 = 81

Where 81 is the number, 3 is the base and 4 is the index or power.

But it can also be said that the logarithm of 81 to base 3 is 4.

i.e.   81 = 34      ……..(1)

and log381 = 4      ………(2)

It is important to note that equations (1) and (2) are equivalent.

34  = 81 is the index form while log381 = 4 is the logarithmic form.

It is necessary to understand how to change from one form to the other. The general form is given as:

If  N = ax, then logaN = x

So we can have the following:

Example 1

Evaluate the following logarithms without using tables or calculator:

1. log28

2. log381

3. log3729

4. log0.25128

5. log80.0625

6. log1.21.728

7. $$\scriptsize \log_{\sqrt [4]{ 2}} \sqrt{512}$$

8. $$\scriptsize \log_{\sqrt [2]{2}}\sqrt{128}$$

Solution:

1. log28

Let log28 = x

changing to index form we have 2x = 8

i.e. 2x = 23

Since we have equal base for both sides then, we can equate the powers

i.e. x = 3.

log28   = 3

2. log381

Let log381 = x

i.e. 3x = 81

i.e.  3x = 34       (equating powers)

i.e.   x = 4.

3. log3729

Let  log3729   = x

3x = 729

i.e.     3x = 35         (equating powers)

i.e. x = 5

4.  log0.25128

Let log0.25128 = x

i.e.     0.25x = 128

i.e. $$\left (\frac{25}{100} \right)^x = \scriptsize 2 ^7$$

i.e. $$\left (\frac{1}{4} \right)^x = \scriptsize 2 ^7$$

:- $$\left (\frac{1}{2^2} \right)^x = \scriptsize 2 ^7$$

2-2x = 27 (equating powers)

-2x = 7

x = $$\scriptsize – \normalsize \frac{7}{2}$$

x = $$\scriptsize -3 \normalsize \frac{1}{2}$$

5. log80.0625

Let log80.0625 = x

i.e. 8x = 0.0625

i.e. $$\frac{625}{10000} = \scriptsize 8 ^x$$

i.e.    $$\scriptsize 8 ^x = \normalsize \frac{1}{16}$$

23x = $$\frac{1}{2^4}$$

23x = 2-4 (equating powers)

i.e. 3x = -4

i.e. $$\scriptsize x = \normalsize \frac{-4}{3} \scriptsize = -1 \normalsize \frac{1}{3}$$

6.  log1.21.728

Let   log1.21.728 = x

i.e.  (1.2)x = 1.728

i.e.  $$\left (\frac{12}{10} \right )^x = \frac{1728}{1000}$$    (Divide Right Hand Side by 8)

i.e. $$\left (\frac{6}{5} \right )^x =\frac{216}{125}$$

i.e. $$\left (\frac{6}{5} \right )^x = \frac{6^3}{5^3} = \left (\frac{6}{5} \right )^3$$

i.e. $$\left (\frac{6}{5} \right )^x = \left (\frac{6}{5} \right )^3$$  (equating powers)

i.e.. x = 3.

7. $$\scriptsize \log_{\sqrt [4]{ 2}} \sqrt{512}$$

Let  $$\scriptsize \log_{\sqrt [4]{ 2}} \sqrt{512} = x$$

i.e.   $$\scriptsize \left ( \sqrt [4]{2} \right)^x = \sqrt{512}$$

i.e. $$\scriptsize \left ( 2^2 \; \times \; 2^{\frac{1}{2}} \right)^x = 512^{\frac{1}{2}}$$

i.e.   (22+1/2)x    =  (29)1/2

i.e.  (25/2)x  = 29/2

i.e. 25x/2 = 29/2      (equating powers)

i.e.  $$\frac {5x}{2} \scriptsize = \normalsize \frac{9}{2}$$

i.e. $$\scriptsize x = \normalsize \frac {9}{5} \scriptsize = 1 \normalsize \frac{4}{5}$$

x = $$\scriptsize 1\normalsize \frac{4}{5}$$

8. $$\scriptsize \log_{\sqrt[2]{2}}\sqrt{128}$$

Let $$\scriptsize \log_{\sqrt[2]{2}}\sqrt{128} = x$$

i.e.$$\left ( \scriptsize \sqrt[2]{2} \right)^x =\scriptsize \sqrt{128}$$

i.e. (21 x 21/2)x  = (128)1/2

i.e.  (21+1/2)x  = (27)1/2

i.e. (23/2)x  = (27/2)

i.e.   23x/2 = 27/2         (equating powers)

i.e. $$\frac {3x}{2} = \frac{7}{2}$$

i.e. x = $$\frac{7}{3} \scriptsize = 2 \normalsize \frac{1}{3}$$

x = $$\scriptsize 2 \normalsize \frac{1}{3}$$

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