4.3 Laws of Logarithms
Logarithms are closely related to equations in index form. When an equation is written in index form as \(N = a^x\), where \(a > 0\) and \(a \neq 1\), it can be converted to the logarithmic form \(\log_a N = x\). Logarithms provide a systematic method to solve exponential equations where the unknowns are in the exponents and cannot be simplified by equating indices directly.
Learning Objectives
- Relate equations in index form to logarithmic form and determine the logarithmic value of a number.
- State and verify the conditions under which logarithms are defined (\(N > 0\), \(a > 0\), and \(a \neq 1\)).
- Relate the graphs of exponential functions and logarithmic functions as inverse functions.
- State and apply the basic laws of logarithms (Product Law, Division Law, and Power Law).
- Prove and apply the change of base formula, including conversion to common logarithms (base 10) and natural logarithms (base \(e\)).
- Solve equations involving indices and logarithms, including real-world applications.
Relationship Between Exponential and Logarithmic Functions and Laws of Logarithms
See how \(y = a^x\) and \(y = \log_a x\) are inverse functions reflected across the line \(y = x\), and review the three fundamental laws of logarithms and the change of base formula.
Index Form
\[N = a^x \quad \text{where } a > 0 \text{ and } a \neq 1\]
Example: \(16 = 2^4\)
Logarithmic Form
\[\log_a N = x \iff N = a^x\]
Example: \(\log_2 16 = 4\)
Product Law
\(\log_a xy = \log_a x + \log_a y\)Division Law
\(\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y\)Power Law
\(\log_a x^n = n \log_a x\)Change of Base Formula
\(\log_a b = \dfrac{\log_c b}{\log_c a}\)Short Explanation
Logarithms are the inverse of exponential expressions.
1. Relationship Between Index Form and Logarithmic Form
An equation in index form is written as \(N = a^x\) where \(a > 0\) and \(a \neq 1\). The equivalent logarithmic form is \(\log_a N = x \iff N = a^x\). The base of the index number is the base of the logarithm, and the power of the index number is the logarithmic value.
2. Special Conclusions
\(a^0 = 1 \iff \log_a 1 = 0\)
\(a^1 = a \iff \log_a a = 1\)
\(\log_a a^x = x\)
3. Restrictions and Existence of Logarithms
\(\log_a N\) is defined only if \(N > 0\), \(a > 0\), and \(a \neq 1\). Logarithms of zero or negative numbers are undefined. Logarithms with a base of 0, 1, or negative numbers are undefined. Base 1 is not used because \(1^n = 1\) for any value of \(n\).
4. Common Logarithms and Antilogarithms
A common logarithm is a logarithm with base 10, written as \(\log_{10} a\) or \(\lg a\). If \(\log_{10} N = x\), then \(N\) is called the antilogarithm of \(x\): \(\text{antilog } x = N \iff 10^x = N\).
5. Graphs of Exponential and Logarithmic Functions
Let \(f(x) = a^x\). Its inverse function is \(f^{-1}(x) = \log_a x\). The graphs of \(y = a^x\) and \(y = \log_a x\) are reflections of each other in the line \(y = x\). \(y = a^x\) passes through \((0, 1)\), while \(y = \log_a x\) passes through \((1, 0)\).
6. Basic Laws of Logarithms
If \(a\), \(x\), and \(y\) are positive and \(a \neq 1\):
Product Law: \(\log_a xy = \log_a x + \log_a y\)
Division Law: \(\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y\)
Power Law: \(\log_a x^n = n \log_a x\) for any real number \(n\).
7. Change of Base Formula
If \(a\), \(b\), and \(c\) are positive numbers, \(a \neq 1\) and \(c \neq 1\): \(\log_a b = \dfrac{\log_c b}{\log_c a}\). Special case when \(c = b\): \(\log_a b = \dfrac{1}{\log_b a}\).
8. Natural Logarithms
A natural logarithm is a logarithm with base \(e\), where \(e\) is an irrational non-recurring decimal (\(e \approx 2.7182\dots\)). Denoted as \(\log_e x\) or \(\ln x\). Properties: \(\ln e = 1\), \(\ln e^x = x\), \(e^{\ln x} = x\), and \(\log_a b = \dfrac{\ln b}{\ln a}\).
| Law / Identity | Expression | Condition |
|---|---|---|
| Logarithm of 1 | \(\log_a 1 = 0\) | \(a > 0, a \neq 1\) |
| Logarithm of Base | \(\log_a a = 1\) | \(a > 0, a \neq 1\) |
| Product Law | \(\log_a xy = \log_a x + \log_a y\) | \(x, y > 0\) |
| Division Law | \(\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y\) | \(x, y > 0\) |
| Power Law | \(\log_a x^n = n \log_a x\) | \(x > 0\), \(n\) real |
| Change of Base | \(\log_a b = \dfrac{\log_c b}{\log_c a}\) | \(a, b, c > 0\), \(a \neq 1\), \(c \neq 1\) |
Steps for Solving Equations with Different Bases
Take common logarithms (\(\log_{10}\)) or natural logarithms (\(\ln\)) on both sides.
Expand using the power law.
Group variable terms on one side.
Solve for \(x\).
Common Logarithm
\(\log_{10} 10 = 1\)\(10^{\log x} = x\)
Natural Logarithm
\(\ln e = 1\)\(e^{\ln x} = x\)
Try to Answer First
Answer in your mind, then press "Check Answer".
Convert the index equation \(5^3 = 125\) into logarithmic form, and solve for \(x\) in \(\log_2 x = 5\).
Express \(2\log_3 x + \log_3 y - 1\) as a single logarithm.
Solve the equation \(4^{2x - 1} = 7^x\), giving the answer correct to three decimal places.
Common Mistakes
- Writing \(\log_a (x + y)\) as \(\log_a x + \log_a y\). The Product Law states that \(\log_a (xy) = \log_a x + \log_a y\), not the sum of terms inside the argument.
- Confusing \(\dfrac{\log_a x}{\log_a y}\) with \(\log_a \left(\dfrac{x}{y}\right)\). The quotient \(\dfrac{\log_a x}{\log_a y}\) is a change of base expression (\(\log_y x\)), whereas \(\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y\).
- Forgetting that \((\log_a x)^n \neq n \log_a x\). The power law applies only when the argument is raised to the power (\(\log_a x^n = n \log_a x\)), not the entire logarithm function.
- Forgetting to convert integer constants into logarithmic form when simplifying expressions (e.g., forgetting that in base \(a\), the number \(1 = \log_a a\) or \(2 = 2\log_a a = \log_a a^2\)).
- Incorrectly attempting to evaluate logarithms of negative numbers or zero (e.g., \(\log_{10}(-5)\) or \(\log_2 0\)), which are undefined.
Concept Misunderstandings
Students sometimes think \(\log_a 1 = 1\) because the input is 1.
Since \(a^0 = 1\) for any \(a > 0\) (\(a \neq 1\)), \(\log_a 1 = 0\). On the other hand, \(\log_a a = 1\) because \(a^1 = a\).
Students assume that the base of a logarithm can be any real number, including negative numbers or 1.
The base \(a\) must strictly be positive and not equal to 1 (\(a > 0, a \neq 1\)). If \(a = 1\), \(1^x = 1\) for all \(x\), so no unique exponent can be determined. If \(a \le 0\), the function does not yield real, continuous values.
Students believe natural logarithm \(\ln x\) follows different rules than regular logarithms.
\(\ln x\) is simply a logarithm with base \(e\) (\(\log_e x\)). It obeys all standard laws of logarithms (Product, Division, and Power Laws) identically.
Summary
- Equivalence Definition: \(\log_a N = x \iff N = a^x\) where \(a > 0, a \neq 1\), and \(N > 0\).
- Fundamental Identities: \(\log_a 1 = 0\), \(\log_a a = 1\), and \(\log_a a^x = x\).
- Laws of Logarithms: Product Law: \(\log_a xy = \log_a x + \log_a y\); Division Law: \(\log_a \left(\dfrac{x}{y}\right) = \log_a x - \log_a y\); Power Law: \(\log_a x^n = n \log_a x\).
- Change of Base: \(\log_a b = \dfrac{\log_c b}{\log_c a}\) and \(\log_a b = \dfrac{1}{\log_b a}\).
- Special Bases: Common logarithm has base 10 (\(\lg x = \log_{10} x\)); Natural logarithm has base \(e\) (\(\ln x = \log_e x\)).
Short Activity
Objective: Test your understanding of logarithmic laws, identities, and the change of base formula.
A. Objective Quiz
1 Which of the following expressions is undefined according to the conditions of logarithms?
2 Given that \(\log_a x = 3\) and \(\log_a y = 2\), what is the value of \(\log_a \left(\dfrac{x^2}{y}\right)\)?
B. Fill in the Blanks
3 If \(\log_{10} N = x\), then the value of \(N\) in terms of antilogarithm is expressed as \(N = \text{antilog }\) ________.
4 By using the change of base law, the logarithm \(\log_a b\) can be written in terms of base \(b\) as \(\dfrac{1}{\text{________}}\).
C. Matching / Drag and Drop
Drag each law or identity to the correct algebraic expression. On a phone, tap a term first, then tap the matching box.
