Properties of Logarithms Study Notes

Properties of Logarithms

General Concepts

  1. Inverse Properties

    • If f(u)=auf(u) = a^u, then g(x)=extlogaxg(x) = ext{log}_a x.

    • Therefore, f(g(x))=aextlogax=xf(g(x)) = a^{ ext{log}_a x} = x and g(f(u))=extloga(au)=ug(f(u)) = ext{log}_a (a^u) = u.

Exponential and Logarithmic Functions

  • Exponential Function: f(x)=axf(x) = a^x

  • Logarithmic Function: f(x)=extlogaxf(x) = ext{log}_a x

  • Example Relationships:

    • extlogaax=xext{log}_a a^x = x

    • extloga(ak)=kext{log}_a (a^k) = k

Examples of Evaluating Logarithms

  1. extlog22=1ext{log}_2 2 = 1

  2. extlogeex=xext{log}_{e} e^x = x

  3. extlog10102=2ext{log}_{10} 10^2 = 2

  4. extlog232=5ext{log}_{2} 32 = 5 (since 32=2532 = 2^5)

  5. extlog2(23)=3ext{log}_{2} (2^3) = 3

  6. extlne2=2ext{ln} e^2 = 2

Value Evaluations

  • Evaluate:

    1. extlog2100ext{log}_{2}100

    2. extlog327ext{log}_3 27

    3. extlog101000ext{log}_{10} 1000 (each answer is the exponent to which the base must be raised to produce that number)

More Properties of Logarithms

  1. Addition of Logs:

    • extloga(xy)=extlogax+extlogayext{log}_a(xy) = ext{log}_a x + ext{log}_a y

  2. Subtraction of Logs:

    • extloga(x/y)=extlogax−extlogayext{log}_a(x/y) = ext{log}_a x - ext{log}_a y

  3. Power Property:

    • extloga(xk)=kimesextlogaxext{log}_a(x^k) = k imes ext{log}_a x

Proofs

  • Proof of Logarithm Properties:

    • For extlog(xy)ext{log}(xy)

    • Starting with x=amx = a^{m} and y=any = a^{n}

    • Then, extlog(xy)=extlog(aman)=extlog(am+n)=m+next{log}(xy) = ext{log}(a^{m}a^{n}) = ext{log}(a^{m+n}) = m+n

    • Hence, extlog(x)+extlog(y)=extlog(xy)ext{log}(x) + ext{log}(y) = ext{log}(xy).

Practice Problems

  1. Use properties of logarithms to find values without a calculator:

    • extlog82ext{log}_8 2

    • extlog416ext{log}_4 16 (evaluate as an exercise in properties)

  2. For example, 3extlog223 ext{log}_2 2 could be represented as extlog2(23)ext{log}_2(2^3).

Summarization of Logarithmic Relationships

  • One more property to note:

    • Consider the equation extlog29=xext{log}_2 9 = x which can be rewritten as:

    • 2x=92^x = 9.

    • By taking natural logarithms: xextln2=extln9x ext{ln} 2 = ext{ln} 9 gives:

    • x=racextln9extln2x = rac{ ext{ln} 9}{ ext{ln} 2}.

Change of Base Formula

  • The change of base formula states:
    extlogax=racextlogbxextlogbaext{log}_a x = rac{ ext{log}_b x}{ ext{log}_b a}

  • In particular, for natural logarithms,
    extlogax=racextlnxextlnaext{log}_a x = rac{ ext{ln} x}{ ext{ln} a}.

Evaluating Logarithms (Rounded Answers)

  1. extlog221extroundedtothenearestthousandth=4.392ext{log}_{2} 21 ext{ rounded to the nearest thousandth} = 4.392

  2. extlog99=1ext{log}_{9} 9 = 1

  3. extlog10102=2ext{log}_{10} 10² = 2

  4. Perform iterations and provide rounded values as needed, e.g.,:

    • extlog8(21)<br>ightarrowextevaluatedandrounded=1.439ext{log}_{8}(21) <br>ightarrow ext{evaluated and rounded} = 1.439

Simplification Examples (No Calculator Allowed)

  1. extlog5extlog8ext{log}_5 ext{log}_8

    • Combine using properties to find relationships and outputs.

    • E.g., extlog82=0.333ext{log}_8 2 = 0.333

  2. extlog<em>4extlog</em>81ext{log}<em>4 ext{log}</em>{81}

    • Using the relationship of extlog416ext{log}_4 16 which pertains to values considered.

Additional Practice Assignments

  1. Assignment 33: Simplification tasks to further enhance understanding and competency in logarithmic properties.

  2. More Practice Exercises covering combinations and evaluations of logarithmic relationships.