Evaluating Logarithmic Expressions: Exact Values (Parts a and b)

Key ideas and definitions

  • Logarithm-to-exponential relationship: if $y = \log_a x$, then $a^y = x$.
  • Inverse relationship: $\log_a x$ is the exponent to which base $a$ must be raised to obtain $x$.
  • Inverse function principle used here: if $a^u = a^v$ and the base $a$ is positive and not equal to 1, then $u = v$.
  • Domain restrictions: base $a$ must satisfy $a>0$ and $a \neq 1$; argument $x$ must satisfy $x>0$.

How to find the exact value of a logarithm (exponential form)

  • To find $\loga x$ exactly, set $y = \loga x$.
  • Convert to exponential form: $a^y = x$.
  • If both sides have the same base and are equal, then the exponents are equal: $a^u = a^v \Rightarrow u = v$.
  • This allows you to equate exponents once you express the right-hand side as a power of the same base.

Step-by-step method (summary)

  • Step 1: Let $y = \log_a x$.
  • Step 2: Rewrite as exponential form: $a^y = x$.
  • Step 3: If possible, express $x$ as a power of $a$ (i.e., $x = a^k$) so that $a^y = a^k$.
  • Step 4: Conclude $y = k$.
  • Step 5: State the exact value: $\log_a x = k$.

Example Part a

  • Given: $\log_3 81$.
  • Let $y = \log_3 81$.
  • Exponential form: $3^y = 81$.
  • Express 81 as a power of 3: $81 = 3^4$.
  • So $3^y = 3^4$.
  • Equate exponents: $y = 4$.
  • Conclusion: \log_{3} 81 = 4.

Example Part b

  • Given: $\log_5 \left( \frac{1}{125} \right)$.
  • Let $y = \log_5 \left( \frac{1}{125} \right)$.
  • Exponential form: $5^y = \frac{1}{125}$.
  • Express the right-hand side as a power of 5: ( \frac{1}{125} = 5^{-3} ) (since $125 = 5^3$).
  • So $5^y = 5^{-3}$.
  • Equate exponents: $y = -3$.
  • Conclusion: \log_{5} \left( \frac{1}{125} \right) = -3.

Key reasoning notes

  • The method hinges on recognizing exact power representations of the arguments.
  • If the argument is not a clean power of the base, this direct exponent matching may not be possible; alternative methods (like evaluating via properties or using a calculator) may be needed.
  • The approach is especially straightforward when the argument is a simple reciprocal or a clear integer power of the base.

Quick practice prompts

  • Evaluate $\log_2 8$ by converting to exponential form.
  • Evaluate $\log_{10} 1000$ using the same steps.

Connections to foundational concepts

  • Reinforces the inverse relationship between exponentiation and logarithms.
  • Demonstrates the importance of recognizing powers and bases to equate exponents.
  • Links to the monotonicity of exponential functions for positive bases not equal to 1.

Practical and conceptual implications

  • Exact values simplify algebraic manipulations and solve equations where the variable appears in an exponent.
  • Understanding these steps lays the groundwork for more advanced logarithmic identities and logarithmic equations.

Summary formulas

  • Fundamental equivalence: y = \log_a x \iff a^y = x
  • Exponent equality rule (for $a>0$, $a\neq 1$): if a^u = a^v\,, then u = v.
  • Example results: \log{3} 81 = 4, \quad \log{5} \left( \frac{1}{125} \right) = -3.