Logarithms, Laws and Applications for SS 1

Logarithms

Performance Objectives

By the end of the lesson, students should be able to:

1. define logarithms and antilogarithms;
2. change indices notations to logarithm(log) notations;
3. plot graph of $y = 10^x$;
4. use digital tools to find the log and antilog of a given number.

Definition of Logarithms and Relationship with Indices

A logarithm is the exponent or power to which a specified base must be raised to yield a given number.

An antilogarithm is the inverse operation of a logarithm, representing the original number obtained by raising the base to the power of the logarithmic value.

Logarithms and indices are closely related. If a number is expressed in index form, it can also be written in logarithmic form. The general relationship is defined as follows:

If $b^y = x$, then $\log_b x = y$ (read as “the logarithm of $x$ to base $b$ is equal to $y$”).

In this relationship, the terms are defined as:

1. $b$ is the base, which must be a positive number greater than zero and not equal to $1$.
2. $y$ is the index, exponent, or logarithm.
3. $x$ is the number or argument.

Deducing Logarithms from Indices

To convert an expression from index form to logarithmic form, the base of the index becomes the base of the logarithm, the power becomes the result of the logarithmic expression, and the value becomes the argument of the logarithm.

Examples of conversions from index form to logarithmic form include:

1. Since $10^2 = 100$, it follows that $\log_{10} 100 = 2$.
2. Since $2^3 = 8$, it follows that $\log_2 8 = 3$.
3. Since $5^4 = 625$, it follows that $\log_5 625 = 4$.
4. Since $10^{-1} = 0.1$, it follows that $\log_{10} 0.1 = -1$.

Deducing Logarithms from Standard Form

Every positive number can be written in standard form as $A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer. This standard form representation is directly linked to common logarithms (logarithms to base 10).

For example, if a number $N = 345.6$, its standard form is $3.456 \times 10^2$. Taking the logarithm to base 10 of this number yields:

$$\log_{10} 345.6 = \log_{10} (3.456 \times 10^2)$$

Using the laws of logarithms, this expression can be expanded as:

$$\log_{10} 345.6 = \log_{10} 3.456 + \log_{10} 10^2$$

$$\log_{10} 345.6 = \log_{10} 3.456 + 2$$

The integer part, $2$, is called the characteristic, and it corresponds directly to the power of $10$ when the number is written in standard form. The decimal part, $\log_{10} 3.456 \approx 0.5386$, is called the mantissa, which is always positive and is obtained from logarithm tables or digital tools.

Therefore, $\log_{10} 345.6 = 2.5386$.

The Graph of $y = 10^x$

The function $y = 10^x$ is an exponential function. Plotting this function allows for the visual determination of logarithms and antilogarithms to base 10. Since $y = 10^x$ is equivalent to $\log_{10} y = x$, the graph can be used to find the logarithm of any number $y$ (by finding $x$) or the antilogarithm of any number $x$ (by finding $y$).

Table of Values for $y = 10^x$

To plot the graph, a table of values is constructed for $x$ ranging from $-1.0$ to $1.0$ with intervals of $0.2$. The corresponding values of $y$ are calculated to two decimal places:

The calculated values are presented in the table below:

Plotting and Reading the Graph

The graph is plotted with the independent variable $x$ on the horizontal axis and the dependent variable $y$ on the vertical axis. The scale is chosen to ensure clear readings, such as $2\text{ cm}$ to represent $0.2\text{ units}$ on the $x$-axis, and $2\text{ cm}$ to represent $1\text{ unit}$ on the $y$-axis.

Observe the graph below to understand the curve of the exponential function.

The graph can be used to perform the following operations:

1. Finding the Logarithm of a Number: To find $\log_{10} 3$, locate $3$ on the vertical $y$-axis, move horizontally to intersect the curve, and then move vertically down to read the corresponding value on the $x$-axis. The value is approximately $0.48$.
2. Finding the Antilogarithm of a Number: To find the antilogarithm of $0.6$ (which is $10^{0.6}$), locate $0.6$ on the horizontal $x$-axis, move vertically up to intersect the curve, and then move horizontally to read the corresponding value on the $y$-axis. The value is approximately $4.0$.

Finding Logarithms and Antilogarithms of Numbers Greater than One

For calculations, logarithms are divided into two parts: the characteristic (the integer part) and the mantissa (the decimal part).

Determining the Characteristic

The characteristic of a number greater than $1$ is determined by counting the number of digits before the decimal point and subtracting $1$. Alternatively, it is the power of $10$ when the number is written in standard form.

Examples of characteristics for numbers greater than $1$ include:

1. For $5.28$, there is $1$ digit before the decimal point. The characteristic is $1 – 1 = 0$. In standard form, $5.28 = 5.28 \times 10^0$, so the power is $0$.
2. For $45.3$, there are $2$ digits before the decimal point. The characteristic is $2 – 1 = 1$. In standard form, $45.3 = 4.53 \times 10^1$, so the power is $1$.
3. For $628.4$, there are $3$ digits before the decimal point. The characteristic is $3 – 1 = 2$. In standard form, $628.4 = 6.284 \times 10^2$, so the power is $2$.

Determining the Mantissa

The mantissa is found using four-figure logarithm tables or digital tools. It is always a positive decimal fraction. When using a table, the first two digits of the number are located in the left-hand column, the third digit is located in the top row, and the fourth digit (if present) is located in the mean difference columns on the right.

Finding the Antilogarithm

To find the antilogarithm of a number, the mantissa (decimal part) is looked up in the antilogarithm table to find the corresponding digits. The characteristic (integer part) determines the position of the decimal point in the final answer. The rule is to place the decimal point after $(n + 1)$ digits from the left, where $n$ is the characteristic.

For example, to find the antilogarithm of $2.5386$:

1. Look up $0.53$ under $8$ in the antilogarithm table, which gives $3451$.
2. Look up the difference for $6$ in the same row, which is $5$, and add it to the value: $3451 + 5 = 3456$.
3. Since the characteristic is $2$, the decimal point is placed after $2 + 1 = 3$ digits.
4. The result is $345.6$.

Calculations Involving Numbers Greater than One

Logarithms simplify complex arithmetic operations by converting multiplication to addition, division to subtraction, powers to multiplication, and roots to division.

The fundamental laws of logarithms used in calculations are:

1. Multiplication Law: $\log_{b} (M \times N) = \log_{b} M + \log_{b} N$
2. Division Law: $\log_{b} \left(\frac{M}{N}\right) = \log_{b} M – \log_{b} N$
3. Power Law: $\log_{b} (M^p) = p \log_{b} M$
4. Root Law: $\log_{b} \left(\sqrt[r]{M}\right) = \frac{1}{r} \log_{b} M$

In standard practice, calculations are structured using a two-column table with headers “Number” and “Logarithm”.

Worked Example 1: Multiplication and Division

Evaluate $\frac{34.28 \times 5.674}{8.921}$ using logarithm tables.

Solution:

First, find the logarithm of each number:

1. For $34.28$, the characteristic is $1$. The mantissa for $34$ under $2$ diff $8$ is $0.5350$. Thus, $\log_{10} 34.28 = 1.5350$.
2. For $5.674$, the characteristic is $0$. The mantissa for $56$ under $7$ diff $4$ is $0.7539$. Thus, $\log_{10} 5.674 = 0.7539$.
3. For $8.921$, the characteristic is $0$. The mantissa for $89$ under $2$ diff $1$ is $0.9504$. Thus, $\log_{10} 8.921 = 0.9504$.

The calculation is structured in the table below:

Now, find the antilogarithm of $1.3385$:

1. Look up $0.33$ under $8$ in the antilogarithm table, which gives $2178$.
2. Look up the difference for $5$ in the same row, which is $3$, and add it: $2178 + 3 = 2181$.
3. Since the characteristic is $1$, place the decimal point after $1 + 1 = 2$ digits from the left.
4. The final value is $21.81$.

Therefore, $\frac{34.28 \times 5.674}{8.921} \approx 21.81$.

Worked Example 2: Powers and Roots

Evaluate $\sqrt{\frac{725.6 \times 4.82}{15.3}}$ using logarithm tables.

Solution:

First, find the logarithm of each number:

1. For $725.6$, the characteristic is $2$. The mantissa for $72$ under $5$ diff $6$ is $0.8607$. Thus, $\log_{10} 725.6 = 2.8607$.
2. For $4.82$, the characteristic is $0$. The mantissa for $48$ under $2$ is $0.6830$. Thus, $\log_{10} 4.82 = 0.6830$.
3. For $15.3$, the characteristic is $1$. The mantissa for $15$ under $3$ is $0.1847$. Thus, $\log_{10} 15.3 = 1.1847$.

The calculation is structured in the table below:

Now, find the antilogarithm of $1.1795$:

1. Look up $0.17$ under $9$ in the antilogarithm table, which gives $1510$.
2. Look up the difference for $5$ in the same row, which is $2$, and add it: $1510 + 2 = 1512$.
3. Since the characteristic is $1$, place the decimal point after $1 + 1 = 2$ digits from the left.
4. The final value is $15.12$.

Therefore, $\sqrt{\frac{725.6 \times 4.82}{15.3}} \approx 15.12$.

How to Use Digital Tools to Find Logarithms and Antilogarithms

Digital tools, such as scientific calculators and mobile mathematics applications, provide quick and precise values for logarithms and antilogarithms.

Finding the Logarithm Using a Scientific Calculator

The procedure for finding the common logarithm (base 10) of a number involves the following steps:

1. Turn on the scientific calculator.
2. Locate and press the $\log$ key (this represents the common logarithm, base 10).
3. Input the number (for example, $345.6$).
4. Press the $=$ key to display the result. The screen will display $2.5385737…$, which rounds to $2.5386$.

Finding the Antilogarithm Using a Scientific Calculator

Since the antilogarithm is the inverse of the logarithm, it is calculated using the $10^x$ function on a calculator. The procedure involves the following steps:

1. Turn on the scientific calculator.
2. Locate and press the SHIFT or 2ndF key.
3. Press the $\log$ key. This activates the secondary function, which is $10^x$.
4. Input the logarithmic value (for example, $1.3385$).
5. Press the $=$ key to display the result. The screen will display $21.80218…$, which rounds to $21.80$ or $21.81$ depending on the decimal places required.

Practice Exercises

Deduction and Conversion

1. Convert the following index notations to logarithmic notations:
   1. $3^5 = 243$
   2. $10^4 = 10000$
   3. $2^7 = 128$
2. Convert the following logarithmic notations to index notations:
   1. $\log_{4} 64 = 3$
   2. $\log_{10} 1000 = 3$
   3. $\log_{5} 125 = 3$

Graph Work

1. Using a scale of $2\text{ cm}$ to represent $0.2\text{ units}$ on the $x$-axis and $2\text{ cm}$ to represent $1\text{ unit}$ on the $y$-axis, plot the graph of $y = 10^x$ for $-1.0 \le x \le 1.0$.
2. From the plotted graph, estimate the values of:
   1. $\log_{10} 5$
   2. $\log_{10} 7.5$
   3. The antilogarithm of $0.5$
   4. The antilogarithm of $-0.3$

Logarithm Table Calculations

Evaluate the following expressions using four-figure logarithm tables, showing all steps in a tabular format:

1. $45.23 \times 3.156$
2. $\frac{854.3}{23.45}$
3. $(6.32)^4$
4. $\sqrt[3]{456.8 \times 2.14}$