Exponentials and Logarithms.

Grade 12 Mathematics: Section 1.9 - Exponentials and Logarithms (Manipulation and Applications)

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Lesson Objectives

  • Understand the relationship between exponentials and logarithms as inverse functions.
  • Apply the laws of exponents and laws of logarithms for manipulation and simplification.
  • Recognize and use the natural exponential function \(e^x\) and the natural logarithm \(\ln x\).
  • Solve exponential and logarithmic equations using algebraic manipulation.
  • Apply exponential and logarithmic models to real-world problems: population growth, radioactive decay, compound interest, pH scale, sound intensity.

Introduction to Exponentials and Logarithms

An exponential function is of the form \(y = a^x\) where \(a > 0\) and \(a \neq 1\). A logarithm is the inverse of an exponential: if \(y = a^x\), then \(x = \log_a y\). This inverse relationship allows us to solve for unknowns in exponents, making logarithms essential for modelling growth, decay, and many natural phenomena.

Fundamental Relationship
\(y = a^x \iff \log_a y = x\)
The logarithm answers the question: "To what power must the base \(a\) be raised to get \(y\)?"
Key Definitions:
• Exponential Function: \(f(x) = a^x\) where \(a > 0, a \neq 1\).
• Logarithm (Base a): \(\log_a x\) is the power to which \(a\) must be raised to obtain \(x\).
• Natural Exponential: \(e^x\) where \(e \approx 2.71828\).
• Natural Logarithm: \(\ln x = \log_e x\).
• Common Logarithm: \(\log x = \log_{10} x\) (used in many applications).

Laws of Exponents and Logarithms

These laws are essential for simplifying expressions and solving equations. Note that the laws of logarithms mirror the laws of exponents because logarithms are exponents.

Laws of Exponents (for \(a > 0\)): \[ a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}, \quad a^0 = 1, \quad a^{-n} = \frac{1}{a^n} \]
Laws of Logarithms (for \(a > 0, a \neq 1\), and \(x, y > 0\)): \[ \log_a (xy) = \log_a x + \log_a y, \quad \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y, \quad \log_a (x^k) = k \log_a x \] Also: \(\log_a 1 = 0\), \(\log_a a = 1\), and the change of base formula: \(\log_a x = \frac{\log_b x}{\log_b a}\).
Example 1: Simplifying using Exponent Laws
Problem: Simplify \(3^4 \times 3^5\).
Solution: \(3^{4+5} = 3^9\). Answer: \(3^9\).
Example 2: Simplifying using Logarithm Laws
Problem: Simplify \(\log_2 8 + \log_2 4\).
Solution: \(\log_2 (8 \times 4) = \log_2 32 = 5\) (since \(2^5 = 32\)).
Example 3: Expanding a Logarithm
Problem: Expand \(\log_3 (27x^2)\).
Solution: \(\log_3 27 + \log_3 x^2 = 3 + 2\log_3 x\) (since \(3^3 = 27\)).
Example 4: Combining Logarithms
Problem: Write as a single logarithm: \(2\log_5 x - \log_5 y\).
Solution: \(\log_5 x^2 - \log_5 y = \log_5 \left(\frac{x^2}{y}\right)\).
Example 5: Change of Base Formula
Problem: Evaluate \(\log_4 32\) using base 2.
Solution: \(\log_4 32 = \frac{\log_2 32}{\log_2 4} = \frac{5}{2} = 2.5\). (Check: \(4^{2.5} = 4^{5/2} = (2^2)^{5/2} = 2^5 = 32\).)

Practice for Laws of Exponentials and Logarithms

  1. Simplify \(7^3 \times 7^4\).
  2. Simplify \(\frac{5^6}{5^2}\).
  3. Expand \(\log_2 (8x^3)\).
  4. Write as a single logarithm: \(3\ln x + 2\ln y\).
  5. Evaluate \(\log_9 27\) using the change of base formula.

Natural Exponential and Natural Logarithm

The number \(e \approx 2.71828\) is a special mathematical constant. The natural exponential function \(f(x) = e^x\) and the natural logarithm \(\ln x = \log_e x\) are inverse functions. They are fundamental in calculus, physics, finance, and many scientific models.

Key Properties:
\(\ln e = 1\), \(e^{\ln x} = x\) (for \(x > 0\)), \(\ln e^x = x\)
Also: \(\ln x = y \iff e^y = x\).
Example 1: Simplifying Natural Exponential Expressions
Problem: Simplify \(e^{2\ln 3}\).
Solution: \(e^{2\ln 3} = e^{\ln 3^2} = 3^2 = 9\).
Example 2: Simplifying Natural Logarithm Expressions
Problem: Simplify \(\ln(e^5)\).
Solution: \(\ln e^5 = 5\).
Example 3: Using Inverse Relationship to Solve
Problem: Given that \(\ln x = 2.5\), find \(x\).
Solution: \(x = e^{2.5} \approx 12.18\).

Practice for Natural Exponential and Logarithm

  1. Simplify \(e^{3\ln 2}\).
  2. Simplify \(\ln(e^7)\).
  3. If \(\ln y = 4\), what is \(y\)?
  4. Simplify \(e^{\ln 5 + \ln 2}\).

Solving Exponential and Logarithmic Equations

To solve exponential equations, take logarithms of both sides or rewrite using the same base. To solve logarithmic equations, exponentiate both sides or use the properties of logs to combine terms.

General Strategies:
• For \(a^x = b\): rewrite as \(x = \log_a b\) or take \(\log\) of both sides: \(x \ln a = \ln b\) so \(x = \frac{\ln b}{\ln a}\).
• For \(\log_a x = b\): rewrite as \(x = a^b\).
• For equations like \(\log_a (x) + \log_a (x-3) = 1\): combine logs, then exponentiate.
• Always check for extraneous solutions (arguments of logs must be positive).
Example 1: Solving an Exponential Equation
Problem: Solve \(2^x = 5\).
Solution: Taking logs: \(x \ln 2 = \ln 5\) → \(x = \frac{\ln 5}{\ln 2} \approx 2.3219\).
Alternative: \(x = \log_2 5\).
Example 2: Solving with the Same Base
Problem: Solve \(3^{2x-1} = 27\).
Solution: \(27 = 3^3\), so \(2x - 1 = 3\) → \(2x = 4\) → \(x = 2\).
Example 3: Solving a Logarithmic Equation
Problem: Solve \(\log_3 (x+2) = 4\).
Solution: Exponentiate: \(x+2 = 3^4 = 81\) → \(x = 79\).
Example 4: Solving with Combined Logarithms
Problem: Solve \(\log_2 x + \log_2 (x-4) = 5\).
Step 1: Combine: \(\log_2 (x(x-4)) = 5\).
Step 2: Exponentiate: \(x(x-4) = 2^5 = 32\) → \(x^2 - 4x - 32 = 0\).
Step 3: Factor: \((x-8)(x+4)=0\) → \(x=8\) or \(x=-4\).
Step 4: Check domain: \(x>0\) and \(x>4\), so \(x=8\) is valid; \(x=-4\) is extraneous. Answer: \(x=8\).
Example 5: Exponential with Base e
Problem: Solve \(e^{2x} = 10\).
Solution: Take natural log: \(2x = \ln 10\) → \(x = \frac{\ln 10}{2} \approx 1.1513\).

Practice for Solving Equations

  1. Solve \(5^x = 7\).
  2. Solve \(4^{2x-1} = 64\).
  3. Solve \(\log_5 (3x+1) = 2\).
  4. Solve \(\log x + \log (x-3) = 1\).
  5. Solve \(e^{3x} = 8\).

Real-World Applications (Growth, Decay, Finance)

Exponential and logarithmic functions model many natural phenomena: population growth, radioactive decay, compound interest, pH scale, sound intensity (decibels), earthquake magnitude (Richter scale), and more.

Key Models:
Exponential growth: \(N(t) = N_0 e^{kt}\) (k > 0)
Exponential decay: \(N(t) = N_0 e^{-kt}\) (k > 0)
Compound interest: \(A = P(1 + \frac{r}{n})^{nt}\) and continuous compounding: \(A = Pe^{rt}\)
pH scale: \(\text{pH} = -\log_{10} [H^+]\)
Richter magnitude: \(M = \log_{10} \frac{I}{I_0}\)
Application 1: Population Growth
Problem: A bacterial culture grows from 100 to 300 in 2 hours. Find the growth rate k and the population after 5 hours (using \(N(t) = N_0 e^{kt}\)).
Step 1: \(300 = 100 e^{2k}\) → \(3 = e^{2k}\) → \(2k = \ln 3\) → \(k = \frac{\ln 3}{2} \approx 0.5493\).
Step 2: After 5 hours: \(N(5) = 100 e^{0.5493 \times 5} = 100 e^{2.7465} \approx 100 \times 15.59 = 1559\).
Application 2: Radioactive Decay (Half-Life)
Problem: Carbon-14 has a half-life of 5730 years. If a sample initially has 100 g, how much remains after 1000 years?
Step 1: Model: \(N(t) = N_0 e^{-kt}\). Half-life means \(50 = 100 e^{-k(5730)}\) → \(0.5 = e^{-5730k}\) → \(-5730k = \ln 0.5\) → \(k = \frac{-\ln 0.5}{5730} \approx 0.000121\).
Step 2: \(N(1000) = 100 e^{-0.000121 \times 1000} = 100 e^{-0.121} \approx 100 \times 0.886 = 88.6\) g.
Application 3: Compound Interest
Problem: If ₦10,000 is invested at 6% per annum compounded quarterly, find the amount after 3 years.
Solution: \(A = P(1 + \frac{r}{n})^{nt} = 10000(1 + \frac{0.06}{4})^{4 \times 3} = 10000(1.015)^{12} \approx 10000 \times 1.1956 = ₦11,956.\)
Application 4: Continuous Compounding
Problem: If ₦5,000 is invested at 5% compounded continuously, find the amount after 10 years.
Solution: \(A = Pe^{rt} = 5000 e^{0.05 \times 10} = 5000 e^{0.5} \approx 5000 \times 1.6487 = ₦8,243.50\).
Application 5: pH Scale
Problem: The hydrogen ion concentration of a solution is \(2.5 \times 10^{-6}\) M. Find its pH.
Solution: \(\text{pH} = -\log_{10}(2.5 \times 10^{-6}) = -(\log_{10} 2.5 + \log_{10} 10^{-6}) = -(0.3979 - 6) = -(-5.6021) = 5.60\).
Application 6: Richter Scale (Earthquake Intensity)
Problem: Earthquake A has magnitude 5.0, Earthquake B has magnitude 7.0. How many times more intense is B than A?
Solution: \(M = \log_{10} (I/I_0)\) → \(I/I_0 = 10^M\). For M=5, \(I_A = 10^5 I_0\); for M=7, \(I_B = 10^7 I_0\). Ratio \(\frac{I_B}{I_A} = 10^{7-5} = 10^2 = 100\) times more intense.

Practice for Applications

  1. A population of 500 grows to 800 in 3 years. Find yearly growth rate (continuous).
  2. Iodine-131 has a half-life of 8 days. If you start with 200 g, how much remains after 20 days?
  3. Find the amount if ₦20,000 is invested at 8% compounded monthly for 2 years.
  4. Find the hydrogen ion concentration if pH = 4.2.
  5. How many times more intense is an earthquake of magnitude 6.5 than one of magnitude 4.5?

Cumulative Practice Exercises

  1. Simplify \(\frac{6^5}{6^3}\).
  2. Write as a single logarithm: \(2\log_3 x - 3\log_3 y\).
  3. Solve \(4^x = 32\).
  4. Solve \(\log_2 (x+1) = 3\).
  5. Solve \(\log_5 x + \log_5 (x-4) = 1\).
  6. Solve \(e^{x} = 12\).
  7. Find the annual growth rate (continuous) if a population triples in 10 years.
  8. How much money will be in an account after 5 years if ₦15,000 is invested at 4% compounded quarterly?
  9. A radioactive substance decays from 60 g to 20 g in 15 years. Find the half-life.
  10. If \(\log_{10} 2 \approx 0.3010\) and \(\log_{10} 3 \approx 0.4771\), find \(\log_{10} 6\) without a calculator.
  11. An earthquake has intensity 10,000 times the reference level. Find its Richter magnitude.
  12. Solve \(\ln(2x) = 5\).
  13. Solve \(3^{x+2} = 81\).
  14. Error analysis: A student solved \(\log x + \log (x-1) = 1\) and got \(x=5, x=-4\). Identify the error and give the correct answer.
  15. The pH of a solution is 3.5. Find \([H^+]\).
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Answers

  1. \(6^{2} = 36\)
  2. \(\log_3 \frac{x^2}{y^3}\)
  3. \(4^x = 32\) → \(2^{2x} = 2^5\) → \(2x=5\) → \(x=2.5\)
  4. \(x+1 = 2^3 = 8\) → \(x=7\)
  5. \(\log_5 (x(x-4)) = 1\) → \(x^2 - 4x = 5\) → \(x^2 - 4x -5=0\) → \((x-5)(x+1)=0\) → \(x=5\) (x=-1 invalid)
  6. \(x = \ln 12 \approx 2.4849\)
  7. \(3 = e^{10r}\) → \(10r = \ln 3\) → \(r = \frac{\ln 3}{10} \approx 0.1099 = 10.99\%\)
  8. \(A = 15000(1 + \frac{0.04}{4})^{20} = 15000(1.01)^{20} \approx 15000 \times 1.22019 = ₦18,302.85\)
  9. \(20 = 60 e^{-15k}\) → \(\frac{1}{3} = e^{-15k}\) → \(-15k = \ln(1/3) = -\ln 3\) → \(k = \frac{\ln 3}{15}\). Half-life: \(t = \frac{\ln 2}{k} = \frac{\ln 2}{\ln 3/15} = 15 \frac{\ln 2}{\ln 3} \approx 15 \times 0.6309 \approx 9.46\) years.
  10. \(\log_{10} 6 = \log_{10} (2 \times 3) = 0.3010 + 0.4771 = 0.7781\)
  11. \(M = \log_{10}(10000) = 4\)
  12. \(2x = e^5\) → \(x = \frac{e^5}{2} \approx 74.206\)
  13. \(3^{x+2} = 3^4\) → \(x+2 = 4\) → \(x=2\)
  14. The domain requires \(x>1\). \(x=-4\) is extraneous. Correct answer: \(x=5\).
  15. \([H^+] = 10^{-3.5} \approx 3.16 \times 10^{-4}\) M

Conclusion & Summary

Exponentials and logarithms are inverse functions that allow us to model growth, decay, and many natural processes. The laws of exponents and logarithms provide powerful algebraic tools for simplifying expressions and solving equations. Applications range from population dynamics and radioactive decay to finance (compound interest) and scientific measurements (pH, Richter scale). Mastering these concepts is essential for calculus, physics, chemistry, economics, and engineering.

Key Takeaways:
1. \(y = a^x \iff \log_a y = x\)
2. Laws of logs: \(\log_a(xy) = \log_a x + \log_a y\), \(\log_a(x/y) = \log_a x - \log_a y\), \(\log_a(x^k) = k \log_a x\)
3. Natural logs: \(\ln x = \log_e x\), \(e^{\ln x} = x\), \(\ln e^x = x\)
4. Solving equations: convert between exponential and logarithmic forms, use log properties to combine, and check for domain restrictions (arguments > 0).
5. Models: Exponential growth \(N(t)=N_0 e^{kt}\), decay \(N(t)=N_0 e^{-kt}\), compound interest \(A = P(1+r/n)^{nt}\), continuous compounding \(A = Pe^{rt}\), pH = -log[H⁺], Richter scale \(M = \log(I/I_0)\).

Keep practising these laws and models — they form the foundation of many advanced mathematics and science topics!

Video Resource

Watch this video for a comprehensive review of exponentials and logarithms.

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