IB Mathematics AA SL Exponents and logarithms Study Notes
IB Mathematics AA SL Exponents and logarithms Study Notes Offer a clear explanation of Exponents and logarithms , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Exponents and logarithms
Laws of Exponents (Integer Exponents)
Laws of Exponents (Integer Exponents)
When working with powers (exponents), there are several fundamental laws that apply to integer exponents. These laws simplify expressions involving multiplication, division, and powers of powers.
| Exponent Rules | |
| For \( a \ne 0, \, b \ne 0 \) | |
| Product Rule | $a^x \cdot a^y = a^{x+y}$ |
| Quotient Rule | $\frac{a^x}{a^y} = a^{x-y}$ |
| Power Rule | $(a^x)^y = a^{xy}$ |
| Power of a Product Rule | $(ab)^x = a^x b^x$ |
| Power of a Fraction Rule | $\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$ |
| Zero Exponent | $a^0 = 1$ |
| Negative Exponent | $a^{-x} = \frac{1}{a^x}$ |
| Fractional Exponent | $a^{\frac{x}{y}} = \sqrt[y]{a^x}$ |
Product of Powers:
\( a^m \times a^n = a^{m+n} \)
Multiply powers with the same base by adding the exponents.
Quotient of Powers:
\( \frac{a^m}{a^n} = a^{m-n} \), \( a \neq 0 \)
Divide powers with the same base by subtracting the exponents.
Power of a Power:
\( (a^m)^n = a^{mn} \)
Raise a power to another power by multiplying the exponents.
Power of a Product:
\( (ab)^n = a^n b^n \)
Distribute the exponent to each factor inside the parentheses.
Power of a Quotient:
\( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \), \( b \neq 0 \)
Distribute the exponent to numerator and denominator.
Zero Exponent:
\( a^0 = 1 \), \( a \neq 0 \)
Any nonzero base raised to zero is 1.
Negative Exponent:
\( a^{-n} = \frac{1}{a^n} \), \( a \neq 0 \)
Negative exponent means reciprocal of the positive exponent.
Example
Simplify the following expressions:
- \( 3^4 \times 3^2 \)
- \( \frac{5^7}{5^3} \)
- \( (2^3)^4 \)
- \( (4 \times 5)^3 \)
- \( \left(\frac{6}{2}\right)^2 \)
- \( 7^0 \)
- \( 2^{-3} \)
▶️ Answer/Explanation
- \( 3^{4+2} = 3^6 = 729 \)
- \( 5^{7-3} = 5^4 = 625 \)
- \( 2^{3 \times 4} = 2^{12} = 4096 \)
- \( 4^3 \times 5^3 = 64 \times 125 = 8000 \)
- \( \frac{6^2}{2^2} = \frac{36}{4} = 9 \)
- \( 7^0 = 1 \)
- \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
Introduction to Logarithms
Introduction to Logarithms
A logarithm is the inverse operation of exponentiation. It answers the question: To what power must a base be raised, to get a given number?
Definition: For \( a > 0 \), \( a \neq 1 \), and \( b > 0 \),
\( a^x = b \iff \log_a b = x \)
This means: “\( \log_a b \)” is the power to which you raise the base \( a \) to get \( b \).
Common Logarithm Bases
- Base 10 (Common Logarithm): \( \log_{10} x \) is often written as \( \log x \).
- Base \( e \) (Natural Logarithm): \( \log_e x \) is written as \( \ln x \), where \( e \approx 2.718 \).
Key Facts:
- \( a^x = b \iff \log_a b = x \)
- \( b > 0 \) because logarithms of zero or negative numbers are undefined in real numbers.
- \( \log_e x = \ln x \) (natural logarithm)
Numerical Evaluation Using Technology
Calculators, computers, and spreadsheet software can evaluate logarithms easily using built-in functions:
- Calculator: Use the \( \log \) button for base 10 logs, and \( \ln \) button for natural logs.
- Excel: Use
=LOG(number, base)for any base, or=LOG10(number)for base 10,=LN(number)for natural log.
Example
Evaluate the following logarithms:
- \( \log_{10} 1000 \)
- \( \log_2 32 \)
- \( \ln e^3 \)
- \( \log_5 25 \)
▶️ Answer/Explanation
- \( \log_{10} 1000 = 3 \) because \( 10^3 = 1000 \)
- \( \log_2 32 = 5 \) because \( 2^5 = 32 \)
- \( \ln e^3 = 3 \) because \( \ln e^x = x \)
- \( \log_5 25 = 2 \) because \( 5^2 = 25 \)