Logarithmic Properties: Fresh Take

  • Use the basic properties of logarithms to simplify expressions and solve equations
  • Combine or separate logarithms using the product and quotient rules
  • Use the power rule to simplify logarithms with exponents
  • Break down or combine complicated logarithm expressions into simpler forms
  • Use the change-of-base formula to calculate and simplify logarithms with different bases

Basic Properties of Logarithms

The Main Idea

  • Logarithmic-Exponential Relationship:
    • [latex]\log_b(x) = y \Leftrightarrow b^y = x[/latex], where [latex]b > 0, b \neq 1[/latex]
  • Zero Property:
    • [latex]\log_b(1) = 0[/latex] for any base [latex]b[/latex]
    • Equivalent to [latex]b^0 = 1[/latex]
  • Identity Property:
    • [latex]\log_b(b) = 1[/latex] for any base [latex]b[/latex]
    • Equivalent to [latex]b^1 = b[/latex]
  • Inverse Properties:
    • [latex]b^{\log_b(x)} = x[/latex]
    • [latex]\log_b(b^x) = x[/latex]
  • Domain Restrictions:
    • [latex]b > 0, b \neq 1[/latex] (base restriction)
    • [latex]x > 0[/latex] (argument restriction)

Using the Product Rule for Logarithms

The Main Idea

  • Product Rule Statement:
    • [latex]\log_b(MN) = \log_b(M) + \log_b(N)[/latex], where [latex]b > 0, b \neq 1[/latex]
  • Relation to Exponent Rules:
    • Analogous to [latex]x^a \cdot x^b = x^{a+b}[/latex]
  • Multiple Factors:
    • [latex]\log_b(wxyz) = \log_b(w) + \log_b(x) + \log_b(y) + \log_b(z)[/latex]
Expand [latex]{\mathrm{log}}_{b}\left(8k\right)[/latex].

Using the Quotient Rule for Logarithms

The Main Idea

  • Quotient Rule Statement:
    • [latex]\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)[/latex], where [latex]b > 0, b \neq 1[/latex]
  • Relation to Exponent Rules:
    • Analogous to [latex]\frac{x^a}{x^b} = x^{a-b}[/latex]
  • Combination with Product Rule:
    • Often used in conjunction with the product rule for full expansion
Expand [latex]{\mathrm{log}}_{3}\left(\frac{7{x}^{2}+21x}{7x\left(x - 1\right)\left(x - 2\right)}\right)[/latex].

Using the Power Rule for Logarithms

The Main Idea

  • Power Rule Statement:
    • [latex]\log_b(M^n) = n \log_b(M)[/latex], where [latex]b > 0, b \neq 1[/latex]
  • Derivation from Product Rule:
    • [latex]\log_b(x^2) = \log_b(x \cdot x) = \log_b(x) + \log_b(x) = 2\log_b(x)[/latex]
  • Application to Roots:
    • [latex]\log_b(\sqrt{x}) = \log_b(x^{\frac{1}{2}}) = \frac{1}{2}\log_b(x)[/latex]
Rewrite [latex]\mathrm{ln}{x}^{2}[/latex].

Rewrite [latex]\mathrm{ln}\left(\frac{1}{{x}^{2}}\right)[/latex].

You can view the transcript for “The Properties of Logarithms” here (opens in new window).

You can view the transcript for “Ex 1: Evaluate a Natural Logarithmic Expression Using the Properties of Logarithms” here (opens in new window).

Expanding and Condensing Logarithms

The Main Idea

  • Expanding Logarithms:
    • Apply rules to break down complex logarithmic expressions
    • Can only expand products, quotients, powers, and roots inside the logarithm
  • Condensing Logarithms:
    • Combine multiple logarithms with the same base into a single logarithm
    • Apply rules in reverse order: Power rule first, then product/quotient rules
  • Important Considerations:
    • Logarithms must have the same base to be combined
    • Addition or subtraction inside the logarithm cannot be expanded

Tip for success

  • When expanding and condensing logarithms, keep in mind that there are often more than one or two good ways to reach a good conclusion. The rules for manipulating exponents and logarithms can be combined creatively. You should try a few different ideas for using the rules on complicated expressions to get practice for finding the most efficient path to take in different situations.
  • When working with more complicated examples, write your work down step by step to avoid making incorrect assumptions. It’s okay to try different rules as you practice creativity as long as you use each rule correctly.
Expand [latex]\mathrm{log}\left(\frac{{x}^{2}{y}^{3}}{{z}^{4}}\right)[/latex].

Expand [latex]\mathrm{ln}\left(\sqrt[3]{{x}^{2}}\right)[/latex].

Expand [latex]\mathrm{ln}\left(\frac{\sqrt{\left(x - 1\right){\left(2x+1\right)}^{2}}}{\left({x}^{2}-9\right)}\right)[/latex].

Use the power rule for logs to rewrite [latex]2{\mathrm{log}}_{3}4[/latex] as a single logarithm with a leading coefficient of 1.

Condense [latex]\mathrm{log}3-\mathrm{log}4+\mathrm{log}5-\mathrm{log}6[/latex].

Rewrite [latex]\mathrm{log}\left(5\right)+0.5\mathrm{log}\left(x\right)-\mathrm{log}\left(7x - 1\right)+3\mathrm{log}\left(x - 1\right)[/latex] as a single logarithm.

Condense [latex]4\left(3\mathrm{log}\left(x\right)+\mathrm{log}\left(x+5\right)-\mathrm{log}\left(2x+3\right)\right)[/latex].

You can view the transcript for “Ex 1: Expand Logarithmic Expressions” here (opens in new window).

Using the Change-of-Base Formula for Logarithms

The Main Idea

  • Change-of-Base Formula: For any positive real numbers [latex]M[/latex], [latex]b[/latex], and [latex]n[/latex], where [latex]n \neq 1[/latex] and [latex]b \neq 1[/latex]: [latex]\log_b M = \frac{\log_n M}{\log_n b}[/latex]
  • Common Applications:
    • Using natural logarithms: [latex]\log_b M = \frac{\ln M}{\ln b}[/latex]
    • Using common logarithms: [latex]\log_b M = \frac{\log M}{\log b}[/latex]
  • Purpose:
    • Evaluate logarithms with bases not available on standard calculators
    • Convert between different logarithmic bases
Change [latex]{\mathrm{log}}_{0.5}8[/latex] to a quotient of natural logarithms.

Evaluate [latex]{\mathrm{log}}_{5}\left(100\right)[/latex] using the change-of-base formula.