Laws of logarithms

Example 2

Use the laws of logarithms to evaluate the following:

a. $\log_{4} 32 + \log_{4} 2$

b. $\log_{3} 270 - \log_{3} 10$

c. $\log_{5} \frac{1}{\sqrt[3]{5}}$

d. $\log_{2} 12 + \log_{2} 3 - \log_{2} 9$

e. $\log_{3} \sqrt[4]{\frac{1}{\sqrt{81}}}$

Solution

$\begin{aligned} \text{a.} \quad & \log_4 32 + \log_4 2 = \log_4 (32 \times 2) \\ & = \log_4 64 \\ & = \log_4 4^3 \\ & = 3\log_4 4 \\ & = 3 \cdot 1 \\ & = 3 \\ \text{b.} \quad & \log_3 270 - \log_3 10 = \log_3 \frac{270}{10} \\ & = \log_3 27 \\ & = \log_3 3^3 \\ & = 3\log_3 3 \\ & = 3 \cdot 1 \\ & = 3 \end{aligned}$

c. $\begin{aligned} \log_5 \frac{1}{\sqrt[3]{5}} &= \log_5 (\sqrt[3]{5})^{-1} \\ &= \log_5 (5)^{-\frac{1}{3}} \\ &= -\frac{1}{3} \log_5 5 \\ &= -\frac{1}{3} \cdot 1 \\ &= -\frac{1}{3} \end{aligned}$

d. $\begin{aligned} \log_2 12 + \log_2 3 - \log_2 9 &= \log_2 \left(\frac{12 \times 3}{9}\right) \\ &= \log_2 4 \\ &= \log_2 2^2 \\ &= 2 \log_2 2 \\ &= 2 \cdot 1 \\ &= 2 \end{aligned}$

e. $\begin{aligned} \log_{3} \sqrt[4]{\frac{1}{81}} & = \log_{3} \left(\frac{1}{81}\right)^{\frac{1}{4}} \\ &= \log_{3} (81^{-1})^{\frac{1}{4}} \\ &= \log_{3} (81)^{-\frac{1}{4}} \\ &= \log_{3} (3^4)^{-\frac{1}{4}} \\ &= \log_{3} (3)^{\frac{4 \times -1}{4}} \\ &= \log_{3} 3^{-1} \\ &= -1 \log_{3} 3 \\ &= -1 \cdot 1 \\ &= -1 \end{aligned}$

This is a formula for change of base. For any positive $a$, $b$ ($a$, $b$ ≠ 0) we have

$\log_{b} x = \frac{\log_{a} x}{\log_{a} b}$

Given,

$\log_{4}{8} = \frac{\log{8}}{\log{4}}$

then find a number which is a common base to both 8 and 4

$$\begin{aligned} &= \frac{\log_2 2^3}{\log_2 2^2} \\ &= \frac{3 \log_2 2}{2 \log_2 2} \\ &= \frac{3 \times 1}{2 \times 1} \\ &= \frac{3}{2} \end{aligned}$$

Example 3

Solve.

$\begin{aligned} \log_{9}\sqrt[3]{27} &=\log_{9}(27)^{-\frac{1}{2}} \\ &=-\frac{1}{2}\log_{9}27 \\ &=-\frac{1}{2}\left(\frac{\log_{3}3^{3}}{\log_{3}3^{2}}\right) \\ &=-\frac{1}{2}\left(\frac{3\log_{3}3}{2\log_{3}3}\right) \\ &=-\frac{1}{2}\times\frac{3}{2} \\ &=-\frac{3}{4} \end{aligned}$

Sometimes you will see a logarithm written without a base, like this: $\log(100)$. This usually means that the base is really 10. It is called a common logarithm. It tells us how many times we must multiply 10 to get the desired number. For example:

$\log(1000) = \log_{10}(10^3) = 3\log_{10}(10) = 3$

This means we need to multiply 10 three times to get 1000. Also, we can express a number in scientific notation and then use the laws of logarithms to find its logarithm. For example, let's find $\log(120)$: First, write 120 in scientific notation:

$120 = 1.2 \times 10^2$

Now use the logarithm laws:

$\log(120) = \log(1.2 \times 10^2) = \log(1.2) + \log(10^2)$

$= \log(1.2) + 2\log(10)$

Since $\log(10) = 1$, we get:

$\log(120) = \log(1.2) + 2$

Using a calculator, $\log(1.2) \approx 0.079$, so:

$\log(120) \approx 0.079 + 2 = 2.079$

Here we deal with all 4 operations which are addition, subtraction, multiplication and division. All operations are just as usual operations except division when we are given a negative characteristic. For example;

When you look at characteristic is not divisible by 2. To make it divisible by 2, increase the characteristic by the amount that will make it divisible by given divisor. In our example we have to increase our characteristic which is $1$ by $1$, which gives total of $2$. $2$ is now divisible by 2. So that I may not get out of my question, I have to add 1 to mantissa. Our question will look like this:

$\frac{1+1+1+0.8996}{2} = \frac{2+1.8996}{2} = \frac{2}{2} + \frac{1.8996}{2} = 1 + 0.9498 = 1.9498$

Example 4

Evaluate the following:

a. $2.6173$ $-1.4291$ $\frac{2.0464}{\text{(remember that characteristic is negative)}}$

b. $5.8888$ $\times \quad 3 \quad \text{(remember that characteristic is negative, when you add 2 you were caring you left with } -13)$ $\frac{13.6664}{}$

Introduction to Logarithmic Tables

Logarithmic tables are mathematical tools used to simplify complex calculations, such as multiplication and division, by converting them into addition and subtraction. Most logarithmic tables are based on base 10, known as common logarithms. These tables consist of logarithms of numbers between 0 and 1, which correspond to numbers between 1 and 10 in their original form.

Reading Logarithms from a Logarithmic Table

To find the logarithm of a number using a logarithmic table, follow these steps:

1. Check the Range: Ensure the number is between 0 and 10 (exclusive). If the number is greater than 10, express it in terms of a number between 0 and 10 by factoring out powers of 10.
2. Locate the Number: Ignore the decimal point and find the first two digits of the number in the leftmost column of the table.
3. Find the Mantissa: Slide your finger along the row to the right to find the column corresponding to the next digit of the number.
4. Read the Logarithm: The intersection of the row and column gives the mantissa (the decimal part) of the logarithm.
5. Add the Characteristic: The characteristic is the power of 10 in the original number. For numbers between 0 and 1, the characteristic is negative; for numbers between 1 and 10, it is 0; for numbers between 10 and 100, it is 1; and so on.

Example 1

Find the logarithm of 5.25 from the table.

Solution:

1. Since 5.25 is between 1 and 10, the characteristic is 0.
2. Ignore the decimal point and look for the number 52 in the leftmost column.
3. Slide your finger along the row of 52 to the right to find the column labeled 5.
4. The intersection of the row 52 and column 5 gives the mantissa 0.7202.
5. The logarithm of 5.25 is 0 + 0.7202 = 0.7202.

Therefore, log 5.25 = 0.7202.

Example 2

Find the logarithm of 15.27 from the table.

Solution:

1. Since 15.27 is between 10 and 100, the characteristic is 1.
2. Ignore the decimal point and look for the number 15 in the leftmost column.
3. Slide your finger along the row of 15 to the right to find the column labeled 2.
4. The intersection of the row 15 and column 2 gives the mantissa 0.1818.
5. Next, use the mean difference table for the next digit (7). In the mean difference table, find the intersection of the row 15 and the column 7, which gives 20.
6. Add the mantissa and the mean difference: 0.1818 + 0.0020 = 0.1838.
7. The logarithm of 15.27 is 1 + 0.1838 = 1.1838.

Therefore, log 15.27 = 1.1838.

Finding the ant-logarithm (inverse logarithm)

To find the number whose logarithm is known, follow these steps:

1. Understand the Ant-Log Table: The ant-log table is used to find the number when the logarithm is known.
2. Write Down the Characteristic: This is the number before the decimal point in the logarithm. It indicates the order of magnitude of the number.
3. Find the Mantissa: Locate the mantissa in the central part of the log table.
4. Read the Number: The intersection of the row and column for the mantissa gives the first part of the number.
5. Add the Mean Difference: If applicable, add the mean difference for the next digit of the mantissa.
6. Place the Decimal Point: Use the characteristic to determine the position of the decimal point.

Example 3

Find the number whose logarithm is 0.7597.

Solution:

1. The characteristic is 0, so the number is between 0 and 10.
2. Locate the mantissa 7597 in the log table. It is found at the intersection of the row labeled 57 and the column 5.
3. The number corresponding to the mantissa 7597 is 575.
4. Since the characteristic is 0, the decimal point is placed to make the number between 0 and 10.
5. Therefore, the number is 5.75.

Thus, antilog 0.7597 = 5.75.

Example 4

Find the number whose logarithm is 2.8699.

Solution:

1. The characteristic is 2, so the number is between 100 and 1000.
2. Locate the mantissa 8699 in the log table. It is found at the intersection of the row labeled .86 and the column 9.
3. The number corresponding to the mantissa 8699 is 7396.
4. Using the mean difference table, the intersection of the row .86 and the column 9 gives 15.
5. Add the mantissa and the mean difference: 7396 + 15 = 7411.
6. Since the characteristic is 2, the decimal point is placed after 3 digits.
7. Therefore, the number is 741.1.

Thus, antilog 2.8699 = 741.1.

Finding Products Using Logarithms

To find the product of two numbers using logarithms, apply the logarithmic identity:

$\log(a \times b) = \log a + \log b$

Example 5

Find the product of 25.75 × 450.

Solution:

1. Let $x = 25.75 \times 450$.
2. Take the logarithm of both sides: $\log x = \log(25.75 \times 450)$.
3. Using the logarithmic identity: $\log x = \log 25.75 + \log 450$.
4. From the logarithmic table: $\log 25.75 = 1.4108$ and $\log 450 = 2.6532$.
5. Add the logarithms: $\log x = 1.4108 + 2.6532 = 4.0640$.
6. To find $x$, take the antilogarithm of 4.0640: $x = \text{antilog}(4.0640)$.
7. From the antilog table, $\text{antilog}(4.0640) = 11590$.

Therefore, $25.75 \times 450 = 11590$.

Always the logarithmic calculations are set out in tabular form to make the solution not too long as above. If we set our example in tabular form it will look like this:

Number	log
25.57 = 2.5575 × 10¹	1.4108
450 = 4.50 × 10²	+2.6532
1.159 = 11590	4.0640

Logarithmic Tables to Find Roots and Power of Numbers

Example 6

Calculate using logarithms

$\sqrt[3]{\frac{318 \times 4344}{\sqrt{17200}}}$

Solution

$\sqrt[3]{\frac{318 \times 4344}{17200}} = \left(\frac{318 \times 4344}{17200}\right)^{\frac{1}{3}}$

Let $x = \left(\frac{318 \times 4344}{17200}\right)^{\frac{1}{3}}$

introduce log both sides

$\log x = \log \left(\frac{318 \times 4344}{17200}\right)^{\frac{1}{3}}$

$\log x = \frac{1}{3} \left( (\log 318 + \log 4344) - \log 17200 \right)$

$x = \text{ant-log} \left( \frac{1}{3} \left( (\log 318 + \log 4344) - \log 17200 \right) \right)$

Number	log
318 = 3.18 × 10²	2.5024
4344 = 4.344 × 10³	+3.6379
	6.1403
17200 = 1.7200 × 10⁴	-4.2355
	1.9048
4.3142 × 10⁰ = 4.3142	1.9048 ÷ 3 = 0.6349