Exponential and Logarithmic Equations: Fresh Take

  • Solve exponential equations by changing them to have the same base or by using logarithms
  • Use the definition of logarithms and their special one-to-one property to solve equations with logarithms

Exponential Equations

The Main Idea

  • One-to-One Property of Exponential Functions: For any real numbers [latex]b[/latex], [latex]S[/latex], and [latex]T[/latex], where [latex]b > 0, b \neq 1[/latex]: [latex]b^S = b^T[/latex] if and only if [latex]S = T[/latex]
  • Key Techniques:
    • Equations with like bases: Set the exponents equal
    • Equations with unlike bases: Rewrite with a common base
  • Limitations:
    • Not all exponential equations have solutions
    • The range of an exponential function is always positive

Problem-Solving Approach

  1. Identify the base(s) in the equation
  2. If bases are different, rewrite terms to have a common base
  3. Apply rules of exponents to simplify if necessary
  4. Use the one-to-one property to set exponents equal
  5. Solve the resulting equation for the unknown
  6. Check for extraneous solutions or no solution scenarios
Solve [latex]{5}^{2x}={5}^{3x+2}[/latex].

Solve [latex]{5}^{2x}={25}^{3x+2}[/latex].

Solve [latex]{5}^{x}=\sqrt{5}[/latex].

Solve [latex]{2}^{x}=-100[/latex].

You can view the transcript for “Ex 1: Solve a Basic Exponential Equation Using the Definition of a Logarithm” here (opens in new window).

Using Logarithms to Solve Exponential Equations

The Main Idea

  • Property of Logarithmic Equality: For [latex]M > 0, N > 0, b > 0, b \neq 1[/latex]: If [latex]\log_b(M) = \log_b(N)[/latex], then [latex]M = N[/latex]
  • Key Techniques:
    • Apply logarithms to both sides of the equation
    • Use common log (base [latex]10[/latex]) or natural log (base [latex]e[/latex]) as needed
    • Utilize logarithm rules to simplify and solve
  • Special Cases:
    • Equations with base [latex]e[/latex]: Use natural logarithm
    • Watch for extraneous solutions

Problem-Solving Approach

  1. Identify if a common base can be found (if not, proceed with logarithms)
  2. Choose appropriate logarithm (common or natural)
  3. Apply the logarithm to both sides of the equation
  4. Use logarithm rules to simplify
  5. Solve for the unknown
  6. Check for extraneous solutions
Solve [latex]{2}^{x}={3}^{x+1}[/latex].

Solve [latex]3{e}^{0.5t}=11[/latex].

Solve [latex]3+{e}^{2t}=7{e}^{2t}[/latex].

Solve [latex]{e}^{2x}={e}^{x}+2[/latex].

Logarithmic Equations

The Main Idea

  • Logarithmic-Exponential Equivalence: Every logarithmic equation [latex]\log_b(x) = y[/latex] is equivalent to the exponential equation [latex]b^y = x[/latex]
  • Key Techniques:
    • Use logarithm rules to simplify expressions
    • Convert logarithmic equations to exponential form
    • Isolate the logarithmic term before converting to exponential form
  • Important Consideration:
    • Check for extraneous solutions, as logarithms are only defined for positive arguments

Problem-Solving Approach

  1. Simplify the logarithmic expression using log rules if necessary
  2. Isolate the logarithmic term on one side of the equation
  3. Convert the equation to exponential form
  4. Solve the resulting exponential equation
  5. Check the solution(s) in the original equation to avoid extraneous solutions
Solve [latex]6+\mathrm{ln}x=10[/latex].

Solve [latex]2\mathrm{ln}\left(x+1\right)=10[/latex].

Use a graphing calculator to estimate the approximate solution to the logarithmic equation [latex]{2}^{x}=1000[/latex] to 2 decimal places.

You can view the transcript for “Solving Logarithmic Equations” here (opens in new window).

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

The Main Idea

The one-to-one property of logarithms states that for any real numbers [latex]S > 0[/latex], [latex]T > 0[/latex], and any positive real number [latex]b ≠ 1[/latex]:

[latex]\log_b S = \log_b T \text{ if and only if } S = T[/latex]

This property allows us to solve logarithmic equations by equating the arguments when the bases are the same.

Key Techniques

  1. Simplify logarithmic expressions using log rules
  2. Apply the one-to-one property to equate arguments
  3. Solve the resulting equation for the unknown
  4. Check for extraneous solutions

Problem-Solving Approach

  1. Combine like terms using logarithm rules to get the equation in the form [latex]\log_b S = \log_b T[/latex]
  2. Apply the one-to-one property to set [latex]S = T[/latex]
  3. Solve the resulting equation for the unknown
  4. Check the solution(s) in the original equation to avoid extraneous solutions
Solve [latex]\mathrm{ln}\left({x}^{2}\right)=\mathrm{ln}1[/latex].