Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Logarithms
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Logarithms
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Subject: General Mathematics
Class: Senior Secondary 1
Term: 1st Term
Week: 1
Theme: Number and Numeration
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Explain and deduce the relationship between in dices and logarithms define logarithm use the graph of y = 10x for multiplication and division Find logarithm and antilogarithm of numbers greater than one use logarithm tables in calculation use log table to solve problems relating to capital market
The concept of logarithms is directly derived from indices (exponents). An index (or exponent) is the power to which a number (the base) is raised. For example, in the expression $10^2 = 100$, 10 is the base, 2 is the index (exponent), and 100 is the result.
A logarithm answers the question: "To what power must the base be raised to get a certain number?" If we have an index statement: $b^x = N$ The equivalent logarithmic statement is: $x = \log_b N$ Here: $b$ is the base (must be a positive number, not equal to 1). $x$ is the exponent or index, which is the logarithm. $N$ is the number (must be positive).
Example 1: Convert the following index forms to logarithmic forms: a) $10^3 = 1000$ b) $5^2 = 25$ c) $2^4 = 16$ Solution: a) If $10^3 = 1000$, then $3 = \log_{10} 1000$. (The power to which 10 must be raised to get 1000 is 3). b) If $5^2 = 25$, then $2 = \log_5 25$. (The power to which 5 must be raised to get 25 is 2). c) If $2^4 = 16$, then $4 = \log_2 16$. (The power to which 2 must be raised to get 16 is 4).
Example 2: Convert the following logarithmic forms to index forms: a) $\log_3 81 = 4$ b) $\log_{10} 0.1 = -1$ Solution: a) If $\log_3 81 = 4$, then $3^4 = 81$. b) If $\log_{10} 0.1 = -1$, then $10^{-1} = 0.1$. For computations in SS1, common logarithms (base 10) are primarily used. When no base is specified, it is generally assumed to be base 10 (e.g., $\log N$ usually implies $\log_{10} N$). The graph of $y = 10^x$ can be used as a visual aid to understand logarithms and perform approximate calculations. This is essentially how a slide rule works. Steps to plot the graph of $y = 10^x$: Create a table of values: Choose a range of $x$ values (e.g., from -2 to 2) and calculate the corresponding $y = 10^x$ values. | $x$ | -2 | -1 | 0 | 0.5 | 1 | 1.5 | 2 | |-------|-------|-------|-------|-------|-------|--------|-------| | $y=10^x$ | 0.01 | 0.1 | 1 | 3.16 | 10 | 31.6 | 100 | (
Note: $10^{0.5} = \sqrt{10} \approx 3.16$)
Choose appropriate scales: The $y$-axis will need to accommodate a large range (e.g., from 0 to 100 or more), while the $x$-axis might range from -2 to
2. A semi-log graph paper can also be used, where the y-axis is logarithmic, resulting in a straight line. For this topic, using standard graph paper is sufficient. Plot the points and draw a smooth curve. Using the graph for Multiplication ($A \times B$): Recall that $\log(A \times B) = \log A + \log B$. From the graph, find $\log A$ by locating $A$ on the $y$-axis and reading its corresponding $x$-value (which is $\log_{10} A$). Let this be $x_A$. Similarly, find $\log B$ by locating $B$ on the $y$-axis and reading its corresponding $x$-value ($x_B$).
Calculate the sum of the logarithms: $x_{sum} = x_A + x_B$. To find the product $A \times B$, locate $x_{sum}$ on the $x$-axis and read the corresponding $y$-value. This $y$-value is the product. Using the graph for Division ($A \div B$): Recall that $\log(A \div B) = \log A - \log B$. From the graph, find $\log A$ ($x_A$) and $\log B$ ($x_B$). Calculate the difference of the logarithms: $x_{diff} = x_A - x_B$. To find the quotient $A \div B$, locate $x_{diff}$ on the $x$-axis and read the corresponding $y$-value. This $y$-value is the quotient. (Teacher's
Note: Emphasize that graph-reading provides approximations. It helps to illustrate the concept of using logarithms for calculation rather than for precise numerical results, which are obtained from tables or calculators.) Every positive number $N$ can be written in standard form (or scientific notation) as $N = A \times 10^n$, where $1 \le A 1$):** Write the number in standard form ($A \times 10^n$). Determine the characteristic ($n$). This is the exponent of
1
0. Find the mantissa: Look up the number $A$ in the logarithm table. The first two digits of $A$ are found in the left-most column. The third digit is found in the top row (columns 0-9). The fourth digit (if any) is found in the "mean difference" columns (1-9). Add the value from the main table to the value from the mean difference column.
Combine: $\log N = \text{Characteristic} . \text{Mantissa}$.
Example 3: Find $\log 345.6$ Standard form: $345.6 = 3.456 \times 10^2$.
Characteristic: $n = 2$.
Mantissa: Look up '34' under '5' in the log table (gives e.g., 5378). Look under 'mean difference 6' (gives e.g., 8). Mantissa = $0.5378 + 0.0008 = 0.5386$. (Actual values depend on the specific log table used) $\log 345.6 = 2.5386$.
Example 4: Find $\log 78.91$ Standard form: $78.91 = 7.891 \times 10^1$.
Characteristic: $n = 1$.
Mantissa: Look up '78' under '9' (e.g., 8971). Look under 'mean difference 1' (e.g., 1). Mantissa = $0.8971 + 0.0001 = 0.8972$. $\log 78.91 = 1.8972$.
Finding Antilogarithm (Antilog): Antilogarithm is the inverse process of finding a logarithm. If $\log N = x$, then $N = \text{antilog } x$. Given a logarithm $x$, the task is to find the number $N$ it represents. Steps to find Antilog $x$: Separate the characteristic and mantissa: $x = \text{characteristic} . \text{mantissa}$. Use the mantissa to find the digits of the number: Look up the mantissa in the antilogarithm table. Find the mantissa value in the main body of the antilog table. Read the corresponding two digits from the left-most column, the third digit from the top row, and add the mean difference digit. This will give a sequence of digits (e.g., 1234). Use the characteristic to place the decimal point: If the characteristic is $n$, the decimal point is placed such that there are $(n+1)$ digits before the decimal point. Alternatively, multiply the number found in step 2 by $10^{\text{characteristic}}$.
Example 5: Find antilog $2.5386$ Characteristic = 2, Mantissa = 0.
5
3
8
6. Digits from mantissa: Look up '53' under '8' in antilog table (e.g., 3451). Look under 'mean difference 6' (e.g., 5). Digits = $3451 + 5 = 3456$.
Place decimal point: Characteristic is 2, so there should be $(2+1)=3$ digits before the decimal point. Antilog $2.5386 = 345.6$.
Example 6: Find antilog $1.8972$ Characteristic = 1, Mantissa = 0.
8
9
7
2. Digits from mantissa: Look up '89' under '7' (e.g., 7889). Look under 'mean difference 2' (e.g., 4). Digits = $7889 + 4 = 7893$. (Slight difference from Example 4 due to table approximations, should ideally match) Let's assume the table gives 7891 from 0.8971 and then the mean difference makes it 7893 from 0.
8
9
7
2. Place decimal point: Characteristic is 1, so there should be $(1+1)=2$ digits before the decimal point. Antilog $1.8972 = 78.93$. The power of logarithms lies in converting multiplication and division into addition and subtraction, and powers and roots into multiplication and division, respectively. This is based on the fundamental laws of logarithms: For positive numbers $A$ and $B$, and a real number $n$: Product Rule: $\log_{10} (A \times B) = \log_{10} A + \log_{10} B$ Quotient Rule: $\log_{10} (A \div B) = \log_{10} A - \log_{10} B$ Power Rule: $\log_{10} (A^n) = n \log_{10} A$ Root Rule: $\log_{10} (\sqrt[n]{A}) = \log_{10} (A^{1/n}) = \frac{1}{n} \log_{10} A$ Steps for Calculation Using Logarithm Tables: Let the expression to be evaluated be $X$.
Take logarithm of both sides: $\log X = \text{expression involving logs}$. Apply the laws of logarithms to simplify the right-hand side. Find the logarithms of individual numbers using the log table. Perform the addition/subtraction/multiplication/division of the logarithms. Find the antilogarithm of the result to get $X$.
Example 7: Evaluate $45.67 \times 8.912$ using logarithm tables. Let $X = 45.67 \times 8.912$ $\log X = \log (45.67 \times 8.912)$ $\log X = \log 45.67 + \log 8.912$ $\log 45.67$: Characteristic = 1 (from $4.567 \times 10^1$) Mantissa (from table for 4567) = $0.6590 + 0.0007 = 0.6597$ $\log 45.67 = 1.6597$ $\log 8.912$: Characteristic = 0 (from $8.912 \times 10^0$) Mantissa (from table for 8912) = $0.9499 + 0.0001 = 0.9500$ $\log 8.912 = 0.9500$ $\log X = 1.6597 + 0.9500 = 2.6097$ Find Antilog $2.6097$: Characteristic = 2, Mantissa = 0.6097 Digits from antilog table for '609' (e.g., 4064). Mean difference for '7' (e.g., 7). Digits = $4064 + 7 = 4071$.
Place decimal point: 3 digits before decimal (characteristic 2). $X = 407.1$ (approx.)
Example 8: Evaluate $987.6 \div 3.45$ using logarithm tables. Let $X = 987.6 \div 3.45$ $\log X = \log (987.6 \div 3.45)$ $\log X = \log 987.6 - \log 3.45$ $\log 987.6$: Characteristic = 2 (from $9.876 \times 10^2$) Mantissa (from table for 9876) = $0.9943 + 0.0003 = 0.9946$ $\log 987.6 = 2.9946$ $\log 3.45$: Characteristic = 0 (from $3.45 \times 10^0$) Mantissa (from table for 3450) = $0.5378$ $\log 3.45 = 0.5378$ $\log X = 2.9946 - 0.5378 = 2.4568$ Find Antilog $2.4568$: Characteristic = 2, Mantissa = 0.4568 Digits from antilog table for '456' (e.g., 2858). Mean difference for '8' (e.g., 5). Digits = $2858 + 5 = 2863$.
Place decimal point: 3 digits before decimal. $X = 286.3$ (approx.)
Example 9: Evaluate $(15.4)^3$ using logarithm tables. Let $X = (15.4)^3$ $\log X = \log (15.4)^3 = 3 \log 15.4$ $\log 15.4$: Characteristic = 1 (from $1.54 \times 10^1$) Mantissa (from table for 1540) = $0.1875$ $\log 15.4 = 1.1875$ $\log X = 3 \times 1.1875 = 3.5625$ Find Antilog $3.5625$: Characteristic = 3, Mantissa = 0.5625 Digits from antilog table for '562' (e.g., 3648). Mean difference for '5' (e.g., 4). Digits = $3648 + 4 = 3652$.
Place decimal point: 4 digits before decimal. $X = 3652$ (approx.)
Example 10: Evaluate $\sqrt[4]{756.2}$ using logarithm tables. Let $X = \sqrt[4]{756.2} = (756.2)^{1/4}$ $\log X = \log (756.2)^{1/4} = \frac{1}{4} \log 756.2$ $\log 756.2$: Characteristic = 2 (from $7.562 \times 10^2$) Mantissa (from table for 7562) = $0.8785 + 0.0001 = 0.8786$ $\log 756.2 = 2.8786$ $\log X = \frac{1}{4} \times 2.8786 = 0.71965$ (round to 4 decimal places for tables, $0.7197$) Find Antilog $0.7197$: Characteristic = 0, Mantissa = 0.7197 Digits from antilog table for '719' (e.g., 5236). Mean difference for '7' (e.g., 9). Digits = $5236 + 9 = 5245$.
Place decimal point: 1 digit before decimal (characteristic 0). $X = 5.245$ (approx.)
Financial Planning and Investment (Capital Market): As demonstrated in the lesson, logarithms are fundamental in calculating compound interest on savings or loans. Students can appreciate how banks and investors use these calculations to project future values of investments (e.g., retirement funds, startup capital) or to determine repayment schedules for loans (e.g., school fees loans, business expansion loans) offered by Nigerian financial institutions. This directly connects to the financial literacy and entrepreneurship skills crucial for students' future. Population Growth and Environmental Studies: Logarithms are used to model exponential growth or decay, such as population growth in Nigerian cities or the decay of radioactive substances in environmental science. Students can analyze how log scales are used to represent wide ranges of data, making trends clearer. For instance, comparing the growth rates of different Nigerian states or cities over time.
Science and Engineering (Scaling): Many natural phenomena are measured on logarithmic scales. While complex calculations might be beyond SS1, the concept can be introduced: pH Scale in Chemistry: Measures acidity/alkalinity. A small change in pH represents a large change in hydrogen ion concentration (e.g., pH of local soil samples, or water quality).
Sound Intensity (Decibels): Logarithms are used to quantify sound levels. Students can relate this to noise pollution levels in busy Nigerian markets or construction sites.
Richter Scale for Earthquakes: Measures earthquake magnitude. Each unit increase on the Richter scale means a tenfold increase in amplitude. This can be related to seismic activities in certain regions of Nigeria (though not highly seismic, awareness is important).