Anasayfa/ AP Calculus/ Exponential & Logarithmic Functions
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πŸ“ˆ Exponential & Logarithmic Functions

$a^x$, $\log_a x$ β€” properties, domains, and graphs
🎯 In this section you will learn

πŸ“Š Exponential Functions

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$. The base $a$ determines the growth or decay rate.

$$ f(x) = a^x \quad (a > 0, a \neq 1) $$
Domain: $(-\infty, \infty)$ | Range: $(0, \infty)$
Base $a$BehaviorExample
$a > 1$Exponential growth (increasing)$f(x) = 2^x$
$0 < a < 1$Exponential decay (decreasing)$f(x) = (0.5)^x$
Example 1Exponential Growth

Population growth: $P(t) = 1000 \cdot (1.05)^t$, where $t$ is years. After 10 years: $P(10) = 1000 \cdot (1.05)^{10} \approx 1628$.

Example 2Exponential Decay

Radioactive decay: $M(t) = M_0 \cdot (0.5)^{t/h}$, where $h$ is half-life.

πŸ”’ Properties of Exponentials

$$ a^m \cdot a^n = a^{m+n} \quad \frac{a^m}{a^n} = a^{m-n} \quad (a^m)^n = a^{mn} $$ $$ a^0 = 1 \quad a^{-n} = \frac{1}{a^n} \quad a^{1/n} = \sqrt[n]{a} $$

πŸ“ Logarithmic Functions

A logarithmic function is the inverse of an exponential function. $y = \log_a x$ means $a^y = x$, where $a > 0$, $a \neq 1$, and $x > 0$.

$$ y = \log_a x \iff a^y = x $$
Domain: $(0, \infty)$ | Range: $(-\infty, \infty)$
Base $a$NotationExample
$a = 10$$\log_{10} x$ or $\log x$$\log 100 = 2$
$a = e$$\ln x$ (natural log)$\ln e^2 = 2$
Example 3Logarithm Evaluation
β‘ 
$\log_2 8$
$2^3 = 8$, so $\log_2 8 = 3$.
β‘‘
$\log_5 25$
$5^2 = 25$, so $\log_5 25 = 2$.
β‘’
$\log_3 81$
$3^4 = 81$, so $\log_3 81 = 4$.

πŸ“ Properties of Logarithms

$$ \log_a(xy) = \log_a x + \log_a y $$ $$ \log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y $$ $$ \log_a(x^r) = r \cdot \log_a x $$ $$ \log_a a = 1 \quad \log_a 1 = 0 $$
Example 4Expanding Logarithms

Expand $\ln\left(\frac{x^2 y}{z}\right)$.

Solution: $\ln(x^2 y) - \ln z = \ln x^2 + \ln y - \ln z = 2\ln x + \ln y - \ln z$.

Example 5Condensing Logarithms

Condense $3\ln x + 2\ln y - \ln z$ into a single logarithm.

Solution: $\ln x^3 + \ln y^2 - \ln z = \ln\left(\frac{x^3 y^2}{z}\right)$.

πŸ”„ Relationship: Inverse Functions

Exponential and logarithmic functions are inverses of each other:

$$ \log_a(a^x) = x \quad \text{for all } x \in \mathbb{R} $$ $$ a^{\log_a x} = x \quad \text{for } x > 0 $$

🌐 Domain and Range

FunctionDomainRangeKey Feature
$f(x) = a^x$$(-\infty, \infty)$$(0, \infty)$Horizontal asymptote at $y=0$
$f(x) = \log_a x$$(0, \infty)$$(-\infty, \infty)$Vertical asymptote at $x=0$
Example 6Finding Domain

Find the domain of $f(x) = \ln(x-3)$.

Solution: The argument of $\ln$ must be positive: $x-3 > 0 \Rightarrow x > 3$. Domain: $(3, \infty)$.

Example 7Finding Domain

Find the domain of $f(x) = \log_2(5 - x)$.

Solution: $5 - x > 0 \Rightarrow x < 5$. Domain: $(-\infty, 5)$.

πŸ–ΌοΈ Visual Comparison

Interactive graph below. Click buttons to see different exponential and logarithmic functions:

πŸ” Key Takeaway
Exponential and logarithmic functions are inverses. Their graphs are reflections across $y = x$. Understanding them is essential for solving growth/decay problems and many calculus applications.

πŸ“ Practice Problems

Problem 1Evaluate

Evaluate $\log_3 81$.

Show solution

$3^4 = 81$, so $\log_3 81 = 4$.

Problem 2Expand

Expand $\ln(x^4 y^2)$.

Show solution

$\ln(x^4 y^2) = \ln x^4 + \ln y^2 = 4\ln x + 2\ln y$.

Problem 3Domain

Find the domain of $f(x) = \ln(2x + 4)$.

Show solution

$2x + 4 > 0 \Rightarrow 2x > -4 \Rightarrow x > -2$. Domain: $(-2, \infty)$.

Problem 4Condense

Condense $2\ln x - \ln y$ into a single logarithm.

Show solution

$2\ln x - \ln y = \ln x^2 - \ln y = \ln\left(\frac{x^2}{y}\right)$.

Problem 5Inverse Relationship

Simplify $10^{\log_{10} 7}$.

Show solution

$10^{\log_{10} 7} = 7$ (since exponentials and logs are inverses).

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