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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:

PowersLogarithms
1000 = 103log10 1000 = 3
100 = 102 log10 100 = 2
10 = 101log10 10 = 1
1 = 100log10 1 = 0
0.1 = 10-1 log10 0.1 = -1
0.01 = 10-2 log100.01 = -2

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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