IB DP Maths: AA SL复习笔记1.2.3 Solving Exponential Equations

What are exponential equations?

• An exponential equation is an equation where the unknown is a power
  • In simple cases the solution can be spotted without the use of a calculator
  • For example,

• In more complicated cases the laws of logarithms should be used to solve exponential equations

• The change of base law can be used to solve some exponential equations without a calculator
  • For example,

How do we use logarithms to solve exponential equations?

• An exponential equation can be solved by taking logarithms of both sides
• The laws of indices may be needed to rewrite the equation first
• The laws of logarithms can then be used to solve the equation
  • ln (log_e) is often used
  • The answer is often written in terms of ln
• A question my ask you to give your answer in a particular form
• Follow these steps to solve exponential equations
  • STEP 1: Take logarithms of both sides
  • STEP 2: Use the laws of logarithms to remove the powers
  • STEP 3: Rearrange to isolate x
  • STEP 4: Use logarithms to solve for x

What about hidden quadratics?

• Look for hidden squared terms that could be changed to form a quadratic
  • In particular look out for terms such as
    • 4^x = (2²)^x = 2^2x = (2^x)²
    • e ^2x = (e²)^x = (e^x)²