exponents

Exponent Rules

• Context from transcript: When multiplying like bases, you add the exponents. This is the Product Rule for exponents and a foundational idea for simplifying expressions.

• Product rule (same base):

  • Formula: $a^b \cdot a^c = a^{b+c}$

  • Example: $2^3 \cdot 2^5 = 2^{8} = 256$

  • Intuition: The base stays the same; exponents combine to reflect repeated multiplication

• Other core exponent rules (to build on the product rule):

  • Quotient rule (same base): $\frac{a^b}{a^c} = a^{b-c}$

  • Power rule (power of a power): $(a^b)^c = a^{bc}$

  • Product of bases to a power: $(ab)^c = a^c b^c$

  • Zero exponent: $a^0 = 1 \quad (a \neq 0)$

  • Negative exponent: $a^{-n} = \frac{1}{a^n}$

  • Domain note: Real exponents with real values require the base to be positive when not using integers. For integer exponents, negative bases can be allowed in some cases, but stability and real-valued results depend on context.

  • Undefined case: $0^0$ is indeterminate in many contexts.

• Quick practice problems

  • Evaluate: $3^4 \cdot 3^{-2} = 3^{2} = 9$

  • Evaluate: $5^0 = 1$

  • Convert to a single exponent: $7^a \cdot 7^b = 7^{a+b}$

• Connection to the next topic: Exponent rules are the backbone of understanding logarithms, since logs are the inverse operations of exponentiation.

Logarithms Basics

• Definition (inverse of exponent):

  • For a base $b>0, b\neq 1$ and $x>0$, the logarithm $\log_b x = y$ if and only if $b^y = x$.

  • Domain: $x>0$, $b>0$, and $b\neq 1$.

  • Inverse relationship: $b^{\log<em>b x} = x$ and $\log</em>b (b^x) = x$.

• Change of base formula

  • For any positive base $k>0, k\neq 1$:

  • $\log<em>b x = \frac{\log</em>k x}{\log_k b}$

  • Example: $\log<em>2 8 = \frac{\log</em>{10} 8}{\log<em>{10} 2}$ (or use natural logs: $\log</em>2 x = \dfrac{\ln x}{\ln 2}$)

• Logarithm rules (similar to exponent rules, due to inverse relationship)

  • Product rule: $\log<em>b(xy) = \log</em>b x + \log_b y$

  • Quotient rule: $\log<em>b\left(\frac{x}{y}\right) = \log</em>b x - \log_b y$

  • Power rule: $\log<em>b(x^r) = r \log</em>b x$

• Special bases and notations

  • Natural logarithm: $\ln x = \log_e x$ (base $e\approx 2.71828\dots$)

  • Common logarithm: $\log_{10} x$ is sometimes written simply as $\log x$ in base 10

• Important examples

  • $\log_2 8 = 3$ because $2^3 = 8$

  • $\log_3 1 = 0$ because $3^0 = 1$

  • Change of base: $\log<em>{2} 10 = \dfrac{\log</em>{10} 10}{\log<em>{10} 2} = \dfrac{1}{\log</em>{10} 2}$

• Base constraints recap

  • Base must be positive and not equal to 1: $b>0, \ b \neq 1$

  • Argument of a logarithm must be positive: $x>0$

Exponents and Logarithms: Interconversion

• If exponential form is known, you can switch to logarithmic form and vice versa:

  • If $a^b = c$, then $\log_a c = b$.

  • If $\log_a c = b$, then $a^b = c$.

  • Example: From $2^x = 16$, we get $x = \log_2 16 = 4\;$ because $2^4=16$.

• Practical translational examples

  • Express $3^4 = ?$ in logarithmic form: $\log_3 (3^4) = 4$; shows consistency of the rules.

Real-world connections and applications

• Growth and decay modeling

  • Exponential growth/decay: $N(t) = N_0 e^{kt}$

  • Logs linearize exponential relationships: take log of both sides to solve for time or rate.

• Finance: compound interest

  • Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

  • Logs can be used to solve for time or rate when the equation is not easily isolated.

• Everyday problem-solving reminders

  • Use product rule when multiplying same-base powers.

  • Use logarithm rules to simplify products, quotients, or powers inside logs.

Common pitfalls and tips

• Do not mix bases without applying change of base when needed.

• For real-valued results with exponents, keep the base positive (except for integer exponents where negative bases may be acceptable in specific contexts).

• Remember the domain restrictions for logs and exponents to avoid undefined expressions.

• When solving equations, consider taking logs to bring down exponents and isolate the variable.

Quick practice problems (mixed)

• Compute $2^3 \cdot 2^5$.

  • Answer: $2^{8} = 256$

• Compute $3^{-2}$.

  • Answer: $\frac{1}{3^2} = \frac{1}{9}$

• Translate to logarithmic form: If $7^x = 343$, what is x?$</p><ul><li><p>Answer:$x = \log_7 343 = 3$because$7^3 = 343$</p></li></ul></li><li><p>Evaluate using logs:$\log2 16$and$\log{10} 1000$</p><ul><li><p>Answers:$\log2 16 = 4$,$\log{10} 1000 = 3$</p></li></ul></li><li><p>Quick base-change check:$\log{2} 8 = \dfrac{\log{10} 8}{\log_{10} 2}$$ (approx value ~ 3)

• Conceptual recap

  • Exponents describe repeated multiplication; logs describe repeated division of exponents; together they provide flexible tools for simplifying and solving expressions involving growth, decay, and compound processes.