4.6 Exponential and Logarithmic Equations

Exponential and logarithmic equations are powerful tools for modeling real-world phenomena. They're used to describe everything from population growth to radioactive decay, making them essential in fields like biology, finance, and physics.

These equations are inverses of each other, with unique properties that simplify complex calculations. Understanding their relationship and how to solve them is key to mastering this topic and applying it to practical problems.

Exponential Equations

Solving exponential equations with like bases

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• Set exponents equal when bases are the same simplifies solving process
  • $2^x = 2^{5}$ results in $x = 5$ since bases are both 2
• Exponential expressions with same base but different exponents can be solved by setting exponent expressions equal
  • $3^{2x-1} = 3^{4x+7}$ becomes $2x-1 = 4x+7$, solve for $x$ by isolating variable

Logarithms for exponential equations

• Logarithms used to solve exponential equations with different bases
  • Take logarithm of both sides using same base as one exponential term
  • $2^x = 50$ becomes $x = \log_2(50)$ after taking $\log_2$ of both sides
• Change of base formula converts logarithms between different bases
  • $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ where $a$ is a different base than $b$
  • Useful when solving equations with logarithms of different bases
• Common logarithm (base 10) often used for simplification in calculations

Logarithmic Equations

Definition of logarithms in equations

• Logarithm definition: $\log_b(x) = y$ means $b^y = x$
• Rewrite logarithmic equations as exponential using definition to solve
  • $\log_3(x) = 4$ becomes $3^4 = x$, resulting in $x = 81$
• Applying definition converts equation to solvable exponential form

One-to-one property for logarithmic equations

• Logarithmic functions are one-to-one, $\log_b(x) = \log_b(y)$ implies $x = y$
• One-to-one property used to solve equations with logarithms on both sides
  • $\log(x+1) = \log(2x-3)$ simplifies to $x+1 = 2x-3$
• Solve resulting equation, check for extraneous solutions introduced during solving process

Properties and Characteristics

Exponential and Logarithmic Properties

• Exponential properties:
  • Product rule: $a^x \cdot a^y = a^{x+y}$
  • Quotient rule: $\frac{a^x}{a^y} = a^{x-y}$
  • Power rule: $(a^x)^y = a^{xy}$
• Logarithmic properties:
  • Product rule: $\log_a(xy) = \log_a(x) + \log_a(y)$
  • Quotient rule: $\log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)$
  • Power rule: $\log_a(x^n) = n\log_a(x)$

Domain and Range

• Domain of exponential functions: all real numbers
• Range of exponential functions: positive real numbers (y > 0)
• Domain of logarithmic functions: positive real numbers (x > 0)
• Range of logarithmic functions: all real numbers

Applications in Science

• Half-life in radioactive decay: time required for half of a substance to decay
  • Calculated using exponential decay formula: $A(t) = A_0(\frac{1}{2})^{t/t_{1/2}}$
  • $t_{1/2}$ is the half-life, $A_0$ is initial amount, $A(t)$ is amount at time $t$

Applications

Real-world applications of exponential equations

• Exponential growth and decay model various phenomena
  • Population growth, compound interest, radioactive decay modeled using $A = A_0e^{kt}$
    • $A$ is amount at time $t$, $A_0$ is initial amount, $k$ is growth or decay rate
  • Larger $k$ values indicate faster growth or decay rates
• pH scale measures acidity/alkalinity using logarithms
  • pH defined as negative base-10 logarithm of hydrogen ion concentration $[H^+]$
  • $pH = -\log_{10}[H^+]$, lower pH values indicate higher acidity
• Richter scale quantifies earthquake magnitude logarithmically
  • Each 1-point increase on Richter scale represents 10-fold increase in seismic wave amplitude
  • Richter magnitude $M = \log_{10}(A/A_0)$, $A$ is maximum seismic wave amplitude, $A_0$ is reference amplitude
  • Logarithmic scale compresses wide range of earthquake energies into manageable values