Exponential and Logarithmic Functions: Graphs, Properties, and Applications

Exponential and Logarithmic Functions

Graphs of Exponential Functions

Exponential functions have the general form $y = a b^x$, where $a$ is a constant and $b$ is the base. Their graphs exhibit rapid growth or decay depending on the value of $b$.

• Exponential Growth: If $b > 1$, the function increases rapidly as $x$ increases.

• Exponential Decay: If $0 < b < 1$, the function decreases rapidly as $x$ increases.

• Horizontal Asymptote: The line $y = 0$ is a horizontal asymptote for all exponential functions of this form.

Examples:

• $y = 2^x$ (growth)

• $y = 2^{-x}$ (decay)

• $y = e^x$ (growth, base $e$)

• $y = e^{-x}$ (decay, base $e$)

Graphs of Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and have the general form $y = ext{log}_b(x)$ or $y = ext{ln}(x)$ (natural logarithm, base $e$).

• Vertical Asymptote: The line $x = 0$ is a vertical asymptote.

• Domain: $x > 0$

• Range: All real numbers

Examples:

• $y = ext{log}_2(x)$

• $y = ext{log}_2(-x)$ (reflection over the y-axis)

• $y = 2 ext{log}_2(x)$ (vertical stretch)

• $y = ext{log}_2(x-1)$ (horizontal shift right by 1)

Transformations of Exponential and Logarithmic Functions

Transformations include shifts, reflections, and stretches/compressions. For a function $f(x)$:

• Vertical Shift: $f(x) + k$ shifts up by $k$ units.

• Horizontal Shift: $f(x - h)$ shifts right by $h$ units.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches by $a$ if $a > 1$, compresses if $0 < a < 1$.

Example:

• $y = ext{log}(x-1) + 2$ is a log function shifted right by 1 and up by 2.

Applications of Exponential Functions

Compound Interest

Exponential functions model compound interest, where the amount grows by a fixed percentage over regular intervals.

• Quarterly Compounding: $A = P(1 + \frac{r}{n})^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

• Variables: $A$ = final amount, $P$ = principal, $r$ = annual interest rate (decimal), $n$ = number of compounding periods per year, $t$ = time in years

Example:

• Invest $6000$at $8.25\%$ for $4$ years:

• Quarterly: $A = 6000(1 + 0.0825/4)^{4 \times 4} = 8334.15$

• Continuously: $A = 6000e^{0.0825 \times 4} = 8345.81$

• Conclusion: Continuous compounding yields a slightly greater return.

Exponential and Logarithmic Equations

Converting Between Exponential and Logarithmic Forms

Exponential and logarithmic equations can be rewritten in each other's forms:

• Exponential to Logarithmic: $b^y = x \implies \log_b(x) = y$

• Logarithmic to Exponential: $\log_b(x) = y \implies b^y = x$

Examples:

• $5 = \log_3 32 \implies 3^5 = 32$

• $\log_5 125 = y \implies 5^y = 125$

• $\log_6 64 = 6 \implies 6^6 = 64$

Evaluating Logarithms Without a Calculator

• Use properties of exponents and logarithms to solve for $x$:

• $\log_2 64 = x \implies 2^x = 64 \implies x = 6$

• $\log_3 \sqrt{1/9} = x \implies 3^x = 1/3 \implies x = -1$

• $\log_4 1 = x \implies 4^x = 1 \implies x = 0$

• $\log_{13} 13 = x \implies 13^x = 13 \implies x = 1$

Properties of Logarithms

Logarithm Laws

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^k) = k \log_b(x)$

Examples:

• $\log_{10}(10000x) = \log_{10}(10000) + \log_{10}(x) = 4 + \log_{10}(x)$

• $\log_7(\frac{x}{y}) = \log_7(x) - \log_7(y)$

• $\log_2(x^2y^3) = 2\log_2(x) + 3\log_2(y)$

• $\ln(\sqrt{x}) = \frac{1}{2}\ln(x)$

Expanding and Condensing Logarithmic Expressions

• Expand: Use product, quotient, and power rules to write as a sum/difference of logs.

• Condense: Combine multiple logs into a single logarithm.

Example:

• Expand: $\log_2(\frac{x^2y^3}{w}) = 2\log_2(x) + 3\log_2(y) - \log_2(w)$

• Condense: $2\log_3(x) + \log_3(y) - \log_3(z) = \log_3(\frac{x^2y}{z})$

Change of Base Formula

To evaluate logarithms with any base using a calculator:

• $\log_b(a) = \frac{\ln a}{\ln b} = \frac{\log a}{\log b}$

Example:

• $\log_{19} 11 = \frac{\ln 11}{\ln 19} \approx 0.6855$

• $\log_{63} 7 = \frac{\ln 7}{\ln 63} \approx 0.36493$

Solving Exponential and Logarithmic Equations

Solving Exponential Equations

• Isolate the exponential term.

• Take the logarithm of both sides (any base, but usually natural log or base 10).

• Solve for the variable.

Example:

• $32^x = 8 \implies x \log 32 = \log 8 \implies x = \frac{\log 8}{\log 32} = \frac{3}{5}$

• $e^{2x} = 143 \implies 2x = \ln 143 \implies x = \frac{\ln 143}{2}$

Solving Logarithmic Equations

• Combine logs if necessary.

• Rewrite in exponential form.

• Solve for the variable.

Example:

• $\log_2(x+3) = 6 \implies x+3 = 2^6 = 64 \implies x = 61$

Applications: Population Growth

Exponential functions are used to model population growth. The general form is $P(t) = P_0 a^t$, where $P_0$ is the initial population, $a$ is the growth factor, and $t$ is time.

• Example: $f(x) = 574(1.026)^x$ models population growth, where $x$ is years since a starting point.

Summary Table: Exponential vs. Logarithmic Functions

Property	Exponential Function	Logarithmic Function
General Form	$y = a b^x$	$y = \log_b(x)$
Domain	All real numbers	$x > 0$
Range	$y > 0$ (if $a > 0$)	All real numbers
Asymptote	Horizontal: $y = 0$	Vertical: $x = 0$
Inverse	Logarithmic function	Exponential function