QMS110 Lecture 9

Exponential and Logarithmic Equations

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. These equations are fundamental in modeling growth and decay processes in business and economics.

• Definition: An equation of the form $a^x = a^y$ implies $x = y$ for $a > 0$, $a \neq 1$.

• Solving Guidelines:

  1. Isolate the exponential expression on one side of the equation.

  2. Take the logarithm of both sides, then use the Laws of Logarithms to bring down the exponent.

  3. Solve for the variable.

• Example: Solve $5^{2x-1} = 125$.

  1. Rewrite both sides with the same base: $125 = 5^3$.

  2. Set exponents equal: $2x-1 = 3$.

  3. Solve: $2x = 4 \Rightarrow x = 2$.

Quadratic Exponential Equations

Some exponential equations may be quadratic in form, requiring factoring techniques.

• Example: Solve $e^{x^2} - e^x - 6 = 0$.

  1. Let $y = e^x$, then $y^2 - y - 6 = 0$.

  2. Factor: $(y-3)(y+2) = 0 \Rightarrow y = 3$ or $y = -2$.

  3. Since $e^x > 0$, only $y = 3$ is valid: $x = \ln 3 \approx 1.0986$.

Solving Exponential Equations Using Logarithms

When the bases cannot be made equal, logarithms are used to solve for the variable.

• Example: Solve $3^{x+2} = 7$.

  1. Take logarithms: $\log(3^{x+2}) = \log 7$.

  2. Apply properties: $(x+2)\log 3 = \log 7$.

  3. Solve: $x = \frac{\log 7}{\log 3} - 2 \approx -0.2288$.

Properties of Logarithms

Logarithms are the inverses of exponential functions and have several useful properties for simplifying expressions and solving equations.

• Key Properties:

  • $\log_b 1 = 0$

  • $\log_b b = 1$

  • $\log_b (MN) = \log_b M + \log_b N$

  • $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$

  • $\log_b (M^p) = p \log_b M$

  • If $\log_b M = \log_b N$, then $M = N$

• Logarithmic-Exponential Conversion:

  • Exponential to Logarithmic: $b^y = x \iff \log_b x = y$

  • Logarithmic to Exponential: $\log_b x = y \iff b^y = x$

• Example: $\log_5 25 = 2$ is equivalent to $5^2 = 25$.

Expanding and Simplifying Logarithmic Expressions

Logarithmic expressions can be expanded or condensed using the properties above.

• Example: Expand $\log_b \left(\frac{wx}{yz}\right)$: $\log_b w + \log_b x - \log_b y - \log_b z$

Logarithmic Equations

A logarithmic equation is one in which the variable is inside a logarithm. These equations are solved by isolating the logarithmic term and converting to exponential form.

• Solving Guidelines:

  1. Isolate the logarithmic term.

  2. Convert to exponential form.

  3. Solve for the variable.

• Example: Solve $\log_2 (25-x) = 3$.

  1. Convert: $25-x = 2^3 = 8$.

  2. Solve: $x = 17$.

• Domain Considerations:

  • Logarithmic functions are defined only for positive arguments.

  • Example: $f(x) = \ln(4-x^2)$ is defined for $|x| < 2$.

Applications of Exponential and Logarithmic Equations

Exponential and logarithmic equations are widely used in business for modeling growth, decay, and financial calculations.

• Compound Interest:

  • Formula for compound interest: $A = P(1 + r)^n$

  • Formula for continuous compounding: $A = Pe^{rt}$

• Doubling Time:

  • The time required for an investment to double: $t = \frac{\ln 2}{\ln(1 + r)}$ (for annual compounding)

  • Shorter doubling time implies a higher rate of return.

• Example: How long will it take for $1000 to grow to $3000 at 12% compounded annually? $3000 = 1000(1.12)^t \Rightarrow t = \frac{\ln 3}{\ln 1.12} \approx 9.16$ years.

• Demand Functions:

  • Example: If $p = 3000e^{-0.05x}$ and $p = 149.40$, solve for $x$:

  • $149.40 = 3000e^{-0.05x} \Rightarrow e^{-0.05x} = 0.0498 \Rightarrow -0.05x = \ln 0.0498 \Rightarrow x \approx 60$

Summary Table: Properties of Logarithms

Property	Formula
Product Rule	$\log_b (MN) = \log_b M + \log_b N$
Quotient Rule	$\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$
Power Rule	$\log_b (M^p) = p \log_b M$
Change of Base	$\log_b M = \frac{\log_k M}{\log_k b}$
Inverse Property	$b^{\log_b x} = x$

Additional info:

• These notes cover topics from Business Calculus Chapter 4 (Derivatives of Exponential & Logarithmic Functions) and Chapter 10 (Integrals of Inverse, Exponential, & Logarithmic Functions), as well as applications relevant to business and economics.