Mastering Logarithms: Evaluation, Properties, Equations, and Graphs

Introduction

This article condenses a comprehensive video lesson on logarithms, covering how to evaluate logs, use the change‑of‑base formula, expand and condense logarithmic expressions, solve logarithmic equations, and graph both logarithmic and exponential functions.

1. Evaluating Logarithms

• Definition: (\log_{b}a = c) means (b^{c}=a).
• Basic examples:
• (\log_{2}4 = 2) because (2^{2}=4).
• (\log_{2}8 = 3) because (2^{3}=8).
• (\log_{3}9 = 2), (\log_{4}16 = 2), (\log_{3}27 = 3), (\log_{2}32 = 5).
• Common bases: When no base is shown, it is base 10. Thus (\log 10 = 1), (\log 100 = 2), (\log 1000 = 3), (\log 1{,}000{,}000 = 6).
• Decimals and negatives:
• (\log 0.1 = -1), (\log 0.01 = -2), (\log 0.001 = -3).
• Log of 0 or a negative number is undefined.

2. Change‑of‑Base Formula

The formula allows any logarithm to be expressed with a more convenient base (often 10 or (e)): [\log_{a}b = \frac{\log b}{\log a} = \frac{\ln b}{\ln a}] - Example: (\log_{4}16 = \frac{\log 16}{\log 4}=2). - It also lets us convert a log with any base to a natural log: (\log_{5}x = \frac{\ln x}{\ln 5}).

3. Logarithmic Properties

4. Solving Logarithmic Equations

1. Convert to exponential form: (\log_{b}A = C \Rightarrow b^{C}=A).
2. Isolate the variable using algebraic steps.
3. Check for extraneous solutions – the argument of every log must be positive.

Sample problems - (\log_{2}Y = 5 \Rightarrow Y = 2^{5}=32). - (\log_{x+5}(x-3)=3 \Rightarrow (x+5)^{3}=x-3) → solve the resulting polynomial and discard any root that makes the log argument non‑positive. - When the same base appears on both sides, set the arguments equal: (\log_{2}(x+5)=\log_{2}(3x-9) \Rightarrow x+5=3x-9 \Rightarrow x=7). - For equations with different bases, use the change‑of‑base formula or rewrite both sides with a common base.

5. Graphing Exponential and Logarithmic Functions

• Exponential form: (y = a^{x}+c)
• Horizontal asymptote at (y=c).
• Choose two easy points by setting the exponent to 0 and 1.
• Example: (y = 2^{x-3}+1) → points (3,2) and (4,3).
• Logarithmic form: (y = \log_{b}(x-h)+k)
• Vertical asymptote at (x = h).
• Table of points by setting (x-h = 1,2) etc.
• Example: (y = \log_{2}(x-3)+1) → asymptote (x=3), points (4,1) and (5,2).
• Domain & Range
• Exponential: domain ((‑\infty,\infty)), range ((c,\infty)) if (a>1) (or ((‑\infty,c)) if (0<a<1)).
• Logarithmic: domain ((h,\infty)), range ((‑\infty,\infty)).
• Inverse functions: swapping (x) and (y) reflects the graph across the line (y=x). The inverse of (y=\log_{b}(x-h)+k) is (y = b^{x-k}+h).

6. Common Mistakes & Tips

• Never forget that the argument of a log must be greater than zero.
• When a fraction appears inside a log, the result is negative (e.g., (\log_{4}\frac{1}{16} = -2)).
• Always verify potential solutions against the original domain restrictions to avoid extraneous answers.
• Use the change‑of‑base formula on a calculator: type log(value)/log(base).

7. Quick Reference Sheet

• Evaluating: ask “to what power must the base be raised?”
• Change of base: (\log_{a}b = \frac{\log b}{\log a})
• Product rule: add logs.
• Quotient rule: subtract logs.
• Power rule: bring exponent down.
• Domain: argument > 0.
• Range: exponential → (asymptote, ∞); logarithmic → (‑∞, ∞).
• Inverse: swap (x) and (y); exponential ↔ logarithmic.

8. Practice Problems (Suggested for Self‑Study)

1. Evaluate (\log_{5}125), (\log_{6}36), (\log_{2}64), (\log_{3}81).
2. Solve (\log_{2}(x+6)-\log_{2}(x-8)=3).
3. Graph (y = \log_{3}(2-x)+1) and its inverse.
4. Find the domain of (f(x)=\ln(x^{2}+2x-15)).
5. Convert (5^{\log_{5}y}) to a simple expression.

Working through these examples will reinforce the concepts and prepare you for any test question involving logarithms.

Understanding the fundamental definition, properties, and graphing techniques of logarithms—and knowing how to switch between logarithmic and exponential forms—empowers you to evaluate any log expression, solve equations confidently, and avoid common pitfalls such as undefined arguments or extraneous solutions.