Exponential & Logarithmic Functions
IB Mathematics Analysis & Approaches (SL & HL)
📐 Laws of Exponents
Product Rule:
\[a^m \times a^n = a^{m+n}\]
Quotient Rule:
\[\frac{a^m}{a^n} = a^{m-n}\]
Power of a Power:
\[(a^m)^n = a^{mn}\]
Power of a Product:
\[(ab)^n = a^n b^n\]
Zero Exponent:
\[a^0 = 1 \quad (a \neq 0)\]
Negative Exponent:
\[a^{-n} = \frac{1}{a^n}\]
Fractional Exponent:
\[a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\]
📈 Exponential Functions
General Form:
\[f(x) = a^x \quad (a > 0, a \neq 1)\]
The Natural Exponential Function:
\[f(x) = e^x\]
where \(e \approx 2.718\) (Euler's number)
Conversion Formula (given in formula booklet):
\[a^x = e^{x \ln a}\]
Key Properties:
Domain: \(x \in \mathbb{R}\) (all real numbers)
Range: \(y > 0\) (all positive real numbers)
y-intercept: \((0, 1)\)
Horizontal Asymptote: \(y = 0\) (x-axis)
Behavior: If \(a > 1\): exponential growth; If \(0 < a < 1\): exponential decay
📊 Logarithms - Definition
Definition:
A logarithm is the inverse operation of exponentiation. It answers: "To what power must we raise the base to get a certain number?"
\[y = \log_a(x) \iff a^y = x\]
where \(a > 0\), \(a \neq 1\), and \(x > 0\)
Common Logarithm:
\[\log(x) = \log_{10}(x)\]
Base 10 (often written without the subscript)
Natural Logarithm:
\[\ln(x) = \log_e(x)\]
Base \(e\) (natural logarithm)
🔑 Laws of Logarithms
Product Law:
\[\log_a(xy) = \log_a(x) + \log_a(y)\]
Given in formula booklet
Quotient Law:
\[\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\]
Given in formula booklet
Power Law:
\[\log_a(x^k) = k \log_a(x)\]
Given in formula booklet
Change of Base Formula:
\[\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\]
Given in formula booklet. Often used with \(b = 10\) or \(b = e\)
⭐ Important Logarithm Results
Logarithm of 1:
\[\log_a(1) = 0\]
Because \(a^0 = 1\)
Logarithm of the Base:
\[\log_a(a) = 1\]
Because \(a^1 = a\)
Inverse Properties:
\[a^{\log_a(x)} = x\]
\[\log_a(a^x) = x\]
Natural Logarithm Special Cases:
\[\ln(e) = 1\]
\[e^{\ln(x)} = x\]
\[\ln(e^x) = x\]
Reciprocal Rule:
\[\log_a\left(\frac{1}{x}\right) = -\log_a(x)\]
📉 Logarithmic Functions
General Form:
\[f(x) = \log_a(x) \quad (a > 0, a \neq 1)\]
Conversion Formula (given in formula booklet):
\[\log_a(x) = \frac{\ln(x)}{\ln(a)}\]
Key Properties:
Domain: \(x > 0\) (positive real numbers only)
Range: \(y \in \mathbb{R}\) (all real numbers)
x-intercept: \((1, 0)\)
Vertical Asymptote: \(x = 0\) (y-axis)
🔄 Inverse Relationship
Exponential and Logarithmic Functions are Inverses:
\[f(x) = a^x \quad \text{and} \quad f^{-1}(x) = \log_a(x)\]
\[f(x) = e^x \quad \text{and} \quad f^{-1}(x) = \ln(x)\]
Graph Properties:
• The graphs of \(y = a^x\) and \(y = \log_a(x)\) are reflections of each other in the line \(y = x\)
• The domain of one is the range of the other
• They both pass through the point \((1, 0)\) and \((0, 1)\) respectively
🧮 Solving Equations
Solving Exponential Equations:
To solve \(a^x = b\), take the logarithm of both sides:
\[x = \log_a(b) = \frac{\ln(b)}{\ln(a)}\]
Solving Logarithmic Equations:
To solve \(\log_a(x) = b\), exponentiate both sides:
\[x = a^b\]
Important Note:
Always check that solutions satisfy the domain restrictions. Logarithms are undefined for zero and negative numbers.
💰 Common Applications
Compound Interest (discrete):
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
\(A\) = final amount, \(P\) = principal, \(r\) = rate, \(n\) = compounding periods/year, \(t\) = time
Compound Interest (continuous):
\[A = Pe^{rt}\]
Exponential Growth/Decay:
\[N(t) = N_0 e^{kt}\]
\(k > 0\): growth, \(k < 0\): decay
Half-Life/Doubling Time:
\[t = \frac{\ln(2)}{k}\]
where \(k\) is the growth/decay rate
💡 Exam Tip: The logarithm laws and change of base formula are given in the IB formula booklet. Remember that \(\ln(x)\) is just \(\log_e(x)\) and follows all the same logarithm rules. Always check domain restrictions when solving logarithmic equations. Use your GDC to verify solutions.