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9.9 – Solving Logarithmic Equations

Key Terms

  • No key terms for this section.

Review

Remember…
  • Solving logarithmic equations
    • The first step for solving a logarithmic equation is to use inverse operations (on both sides) to isolate the logarithm.
  • Natural logs
    • The base of the natural logarithm (ln x) is e.
    • We make both sides an exponent, base e, to undo the natural logarithm.

  • Exponents and Logs are Inverses

Alg2B - 9.9 Exp Log Rev

Notes

Common Logarithms Natural Logarithms
  • How to solve log_b x=a
    1. Use the definition of a logarithm to write an equivalent exponential equation.
    2. Solve for x, using a calculator.

Alg2B - 9.9 Log Solve Exp Ex

  • How to Solve ln\; x=a
    1. Use the definition of logarithms to change to an exponential equation.
      • log_e x=a <–> x=e^a
      • ln x=a <–> x=e^a
    2. Use a calculator to solve for x.

Alg2B - 9.9 Ln Solve Exp Ex

 

Logs with Coefficients
  • How to Solve a\bullet log_b x=d
    1. Isolate the log by dividing both sides by the coefficient.
      • a\bullet log_b x=d
      • \frac{a\bullet log_b x}{a}=\frac{d}{a}
      • log_b x=\frac{d}{a}
    2. Use the definition of logarithms to change to an exponential equation.
      • log_b x=a <–> x=b^a
    3. Use a calculator to solve for x.

Alg2B - 9.9 coef a log

Logs with Coefficients and Additional Terms
  • How to solve an equation of the form c+a\bullet log_b x=d
    1. Subtract c from both sides of the equation.
    2. Divide both sides of the equation by a (as was done in the previous example).

Alg2B - 9.9 coef a log plus

 

The Earthquake Problem
  • An earthquake that measures 2 on the Richter scale has about 17.3 kilowatt-hours of energy. How much energy does an earthquake have if it measures 3 on the Richter scale?
    • R(x): magnitute of the earthquake
    • x: energy (in kilowatt-hours) of the earthquake
    • Formula: R(x)=0.67\bullet log(0.37x)+1.46

Alg2B - 9.9 Earthquake

An earthquake that measures 3 on the Richter scale has about 537.4 kilowatt-hours of energy.

That’s a little more than 31 times the amount of energy in an earthquake that measures 2 on the Richter scale.


  • How much energy does an earthquake have if it measures 7 on the Richter scale?

Alg2B - 9.9 Earthquake7

This earthquake has about 29 million times the energy of one that measures 2 on the Richter scale!

Examples

  • Ex 1. Which of the following logarithmic equations is equivalent to the exponential equation below?
  • ln\;x=8
    • Answer: x=e^8
  • Ex 2. What is the solution to the equation below? Round your answer to two decimal places.
  • log_6 x=2.1
    • Answer: x=43.06
  • Ex 3. What is the solution to the equation below? Round your answer to two decimal places.
  • 4\bullet ln\; x=8.6
    • Answer: x=8.58
  • Ex 4. What is the solution to the equation below? Round your answer to two decimal places.
  • 4+9\bullet ln\; x=15.8
    • Answer: x=3.71
  • Ex 5. What is the solution to the equation below? Round your answer to two decimal places.
  • 4+4\bullet log_2 x=14
    • Answer: x=5.66
  • Ex 6. What is the solution to the equation below? Round your answer to two decimal places.
  • log_2 (2x-1)=3
    • Answer: x=\frac{9}{2}

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