Lesson Note
INTRODUCTION
Logarithms are a fundamental mathematical tool that help us solve problems involving very large or very small numbers. They allow us to transform multiplication into addition, division into subtraction, and powers into simple products. In everyday life, logarithms appear in measuring the intensity of earthquakes (Richter scale), the loudness of sounds (decibels), and in the growth of populations or investments (compound interest). By learning logarithms, students gain a powerful technique that is used in science, engineering, economics, and many other fields.
In the Nigerian secondary school curriculum, logarithms are introduced after students have mastered indices and exponents. They provide a bridge between algebraic manipulation and real‑world applications. Understanding how to read logarithm tables, solve logarithmic equations, and interpret the graphs of logarithmic functions equips students with the skills needed to tackle advanced topics in calculus and statistics.
MAIN CONTENT
Definition of Logarithm
Logarithms are the inverses of exponential functions. If a positive number $b$ (called the base) raised to an exponent $x$ gives another positive number $y$, then $x$ is the logarithm of $y$ to the base $b$. In symbols, $b^{x}=y$ implies $x=\log_{b}y$. The base $b$ must be greater than $0$ and not equal to $1$, and $y$ must be positive. The most common bases are $10$ (common logarithm) and $e\approx2.71828$ (natural logarithm), written as $\log$ and $\ln$ respectively.
Example – Finding $\log_{10} 500$
Step 1: Identify the base and the argument.
The base is $10$ and the argument is $500$.
Step 2: Use the definition of logarithm.
We want $x$ such that $10^{x}=500$.
Step 3: Recognize that $500$ lies between $10^{2}=100$ and $10^{3}=1000$.
Hence $2 Step 4: Using a calculator or logarithm table, find $x\approx2.6990$. Answer: $\log_{10} 500\approx2.6990$. Practice Questions Question 1: Compute $\log_{10} 0.01$. Answer: $\log_{10} 0.01=-2$. Question 2: Find $\ln e^{5}$. Answer: $\ln e^{5}=5$. Question 3: Determine $\log_{2} 32$. Answer: $\log_{2} 32=5$. Properties of Logarithms The logarithm function satisfies several useful properties that simplify complex expressions. For any positive numbers $a$, $b$, and $c$ (with $c\neq1$): - Product rule: $\log_{c}(ab)=\log_{c}a+\log_{c}b$. - Quotient rule: $$\log_{c}\!\left(\frac{a}{b}\right)=\log_{c}a-\log_{c}b$$. - Power rule: $\log_{c}(a^{k})=k\,\log_{c}a$. - Change of base: $\log_{c}a=\frac{\log_{d}a}{\log_{d}c}$ for any base $d>0$, $d\neq1$. These properties allow us to break down complicated logarithmic expressions into simpler parts. Example – Simplify $$\log_{10}\!\left(\frac{2^{3}\cdot5^{2}}{3}\right)$$ Step 1: Apply the product rule. $$\log_{10}\!\left(2^{3}\cdot5^{2}\right)=\log_{10}2^{3}+\log_{10}5^{2}$$. Step 2: Use the power rule. $3\log_{10}2+2\log_{10}5$. Step 3: Apply the quotient rule with the denominator $3$. $$\log_{10}\!\left(\frac{2^{3}\cdot5^{2}}{3}\right)=3\log_{10}2+2\log_{10}5-\log_{10}3$$. Step 4: Substitute approximate values: $$\log_{10}2\approx0.3010,\ \log_{10}5\approx0.6990,\ \log_{10}3\approx0.4771$$. Step 5: Compute: $$3(0.3010)+2(0.6990)-0.4771\approx0.9030+1.3980-0.4771\approx1.8239$$. Answer: $$\log_{10}\!\left(\frac{2^{3}\cdot5^{2}}{3}\right)\approx1.8239$$. Practice Questions Question 1: Evaluate $\log_{10}(1000\cdot0.1)$. Answer: $3+(-1)=2$. Question 2: Simplify $\log_{2}\!\left(\frac{8}{4}\right)$. Answer: $\log_{2}2=1$. Question 3: Compute $\log_{10}(10^{4}\cdot10^{-2})$. Answer: $4-2=2$. Logarithm Bases and Change of Base Formula The change of base formula allows us to express a logarithm with any base in terms of logarithms with a preferred base. For any $a>0$, $a\neq1$, and any positive base $b$ ($b\neq1$), This is particularly useful when calculators provide only common logarithms ($\log$) or natural logarithms ($\ln$). By choosing $b=10$ or $b=e$, we can evaluate logarithms of any base. Example – Find $\log_{5} 125$ using common logarithms Step 1: Write the change of base formula with base $10$. $$\log_{5}125=\frac{\log_{10}125}{\log_{10}5}$$. Step 2: Recognize that $125=5^{3}$, so $\log_{10}125=\log_{10}5^{3}=3\log_{10}5$. Step 3: Substitute: $$\log_{5}125=\frac{3\log_{10}5}{\log_{10}5}=3$$. Answer: $\log_{5}125=3$. Practice Questions Question 1: Express $\log_{3}81$ using natural logarithms. Answer: $\log_{3}81=\frac{\ln81}{\ln3}$. Question 2: Compute $\log_{7}49$ using common logarithms. Answer: $\log_{7}49=2$. Question 3: Find $\log_{2}32$ using change of base with base $10$. Answer: $$\log_{2}32=\frac{\log_{10}32}{\log_{10}2}\approx5$$. Solving Logarithmic Equations To solve equations involving logarithms, we typically isolate the logarithmic term and then apply the definition of logarithm to convert it into an exponential equation. Once in exponential form, we solve for the unknown variable and verify that the solution lies within the domain (the argument of any logarithm must be positive). Example – Solve $\log_{2}(x-3)=4$ Step 1: Recognize the base and argument. The base is $2$ and the argument is $x-3$. Step 2: Apply the definition of logarithm. $2^{4}=x-3$. Step 3: Compute $2^{4}=16$. Step 4: Solve for $x$. $x-3=16\implies x=19$. Step 5: Check domain: $x-3=16>0$, so the solution is valid. Answer: $x=19$. Practice Questions Question 1: Solve $\log_{10}(2x)=3$. Answer: $x=500$. Question 2: Find $y$ if $\ln(y/5)=2$. Answer: $y=5e^{2}$. Question 3: Solve $5\log_{10}x=2$. Answer: $x=10^{0.4}\approx2.5119$. Logarithmic and Antilogarithmic Tables Before calculators, students used logarithmic tables to look up the logarithm of a number and its antilogarithm (the number corresponding to a given logarithm). A typical table lists the first few digits of the logarithm of numbers between 1 and 10, and the last digits of the antilogarithm. To read a table, one locates the integer part of the number, then reads the decimal part from the table, and finally multiplies by the appropriate power of $10$. Example – Find the antilogarithm of $0.3010$ Step 1: Recognize that $0.3010$ is the logarithm to base $10$. Step 2: Recall that $\log_{10}2=0.3010$ (from tables). Step 3: Therefore, the antilogarithm of $0.3010$ is $2$. Answer: $10^{0.3010}=2$. Practice Questions Question 1: What is the logarithm of $5$ to base $10$? Answer: $\log_{10}5\approx0.6990$. Question 2: Find the antilogarithm of $1.3010$. Answer: $10^{1.3010}\approx20$. Question 3: Using a table, estimate $\log_{10}7$. Answer: $\log_{10}7\approx0.8451$. Graph of Logarithmic Functions The graph of $y=\log_{b}x$ (with $b>1$) has a characteristic shape: it passes through $(1,0)$, rises slowly as $x$ increases, and approaches negative infinity as $x$ approaches $0$ from the right. The domain is $x>0$, and the range is all real numbers. Changing the base $b$ stretches or compresses the graph horizontally. Example – Sketch $y=\log_{10}x$ Step 1: Identify key points. At $x=1$, $y=0$; at $x=10$, $y=1$; at $x=0.1$, $y=-1$. Step 2: Plot these points on the coordinate plane. Step 3: Draw a smooth curve passing through the points, approaching the y‑axis but never touching it. Answer: The graph is a logarithmic curve that increases slowly and passes through the points (1,0), (10,1), and (0.1,-1). Practice Questions Question 1: What is the domain of $y=\log_{2}x$? Answer: $(0,\infty)$. Question 2: Find the y‑intercept of $y=\log_{10}x$. Answer: $(1,0)$. Question 3: Determine the x‑intercept of $y=\log_{e}x$. Answer: $(1,0)$. Applications of Logarithms Logarithms appear in many real‑world contexts. In geology, the Richter scale measures earthquake magnitude using $\log_{10}$; in acoustics, sound intensity is expressed in decibels, which involve $10\log_{10}$ of a power ratio; in finance, compound interest uses $\ln$ to model continuous growth. Understanding logarithms allows students to interpret these measurements and solve practical problems. Example – Convert a sound intensity ratio to decibels Step 1: Suppose a sound has a power ratio of $I=1000$ relative to a reference level. We want $L=10\log_{10}I$. Step 2: Compute $\log_{10}1000=3$. Step 3: Multiply by $10$. $L=10\times3=30$ decibels. Answer: The sound intensity is $30$ decibels. Practice Questions Question 1: If the magnitude of an earthquake is $5.5$ on the Richter scale, what is the power ratio compared to a magnitude $0$ quake? Answer: $10^{5.5}\approx316227.77$. Question 2: Find the continuous growth rate $r$ if $e^{r}=3$. Answer: $r=\ln3\approx1.0986$. Question 3: Convert $0.01$ to a logarithm base $10$. Answer: $\log_{10}0.01=-2$. Logarithms provide a powerful inverse operation to exponentiation, enabling the simplification of multiplication, division, and power expressions. By mastering the definition, properties, change‑of‑base formula, and solution techniques for logarithmic equations, students gain a versatile tool for algebraic manipulation. Logarithm tables and graphs help students visualize the behavior of logarithmic functions, while applications in science and engineering illustrate the practical relevance of these concepts. Mastery of logarithms lays the groundwork for advanced study in calculus, statistics, and various STEM disciplines. Question: Find $\log_{10} 50$. Question: Solve $\ln(3x)=4$ for $x$. Question: Compute $\log_{2}\!\left(\frac{32}{8}\right)$. Question: What is the antilogarithm of $0.6021$ to base $10$? Question: If $y=10^{x}$, express $x$ in terms of $y$.SUMMARY
EVALUATION