# 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

## 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.

• 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.

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.

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).

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: $R(x)=0.67\bullet log(0.37x)+1.46$

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?

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: $x=\frac{9}{2}$