Logarithms Overview

Overview

This lecture covers logarithms, including evaluating logs, properties, solving equations, graphing logarithmic and exponential functions, finding domains and inverses, and applying the change of base formula.

Evaluating Logarithms

• logₐ(b) asks "a to what power equals b?"
• log₂4 = 2 because 2² = 4; log₂8 = 3; log₃9 = 2; log₄16 = 2; log₃27 = 3; log₂32 = 5.
• log₅125 = 3; log₆36 = 2; log₂64 = 6; log₃81 = 4.
• log₇49 = 2; log₃1 = 0 (log of 1 is always 0).
• log(10) = 1, log(100) = 2, log(1000) = 3, log(1,000,000) = 6 (count zeros).
• log(0.1) = -1, log(0.01) = -2, log(0.001) = -3 (negative when <1).
• log(0) and log(negative value) are undefined; inside log must be >0.

Change of Base Formula

• logₐb = log_b/log_a (can use any base, including e for natural logs).
• Example: log₄16 = log(16)/log(4) = 2.

Properties of Logarithms

• logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx - logₐy.
• logₐ(xⁿ) = n·logₐx (exponent moves to front).

Expanding and Condensing Log Expressions

• Combine: log x + log y - log z = log(xy/z).
• Expand: log(x²y⁵/z⁶) = 2log x + 5log y − 6log z.
• Move coefficients to exponents before combining/expanding.

Logarithmic and Exponential Equation Solving

• Convert between log and exponential form: logₐb = c ↔ a^c = b.
• For equations, solve for the variable and check for extraneous solutions (cannot have negatives inside log).
• Natural log (ln) base is e; ln e = 1, ln(1) = 0, ln(e⁵) = 5.

Graphing Logarithmic and Exponential Functions

• Exponential functions: f(x) = a^(x−h) + k, horizontal asymptote at y = k, domain all real numbers, range above asymptote.
• Logarithmic functions: f(x) = logₐ(x−h) + k, vertical asymptote at x = h, domain x > h, range (−∞, ∞).
• Inverse functions: log and exponential functions are inverses; their graphs are symmetrical about y = x.

Domains of Logarithmic Functions

• Set inside of log > 0 to find domain.
• For quadratics, factor and use a sign chart to determine valid intervals.

Finding Inverse Functions

• Replace f(x) with y, swap x and y, solve for y.

Key Terms & Definitions

• Logarithm (logₐb) — The power a must be raised to in order to get b.
• Change of Base Formula — logₐb = log b / log a.
• Natural Logarithm (ln) — Logarithm with base e (~2.718).
• Exponent Rule — logₐ(xⁿ) = n·logₐx.
• Vertical Asymptote — x-value where a log function is undefined.
• Horizontal Asymptote — y-value an exponential function approaches.

Action Items / Next Steps

• Practice evaluating logs and converting between log and exponential forms.
• Use the change of base formula on logs without obvious answers.
• Condense and expand log expressions using properties.
• Solve logarithmic and exponential equations, checking for extraneous solutions.
• Practice graphing basic and shifted log/exponential functions.
• Find domains and inverses for given functions.
• Review any assigned exercises or textbook readings on logarithms and exponents.