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Solving Equations with Log Terms and Other Terms
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Solving Equations with Log Terms and Other Terms
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Solving Equations with Log Terms and Other Terms

Here’s how to solve an equation that contains more than one log and a constant term.

Procedure — To Solve an Equation With Log Terms and Other Terms

Step 1 Rewrite the equation with the logs on one side and the constant term on the other side.

Step 2 Combine the logs into a single log.

Step 3 Convert the equation to exponential form and solve.

 

Example 1

Solve: log6 x + log6 (x - 5) = 2

Solution

Step 1 Rewrite the equation with the logs on one side and the constant term on the other side.

The logs are already on one side of the equation.

 log6 x + log6(x - 5) = 2
Step 2 Combine the logs into a single log.

Use the Log of a Product Property.

 log6 x(x - 5) = 2
Step 3 Convert the equation to exponential form and solve.

Convert to exponential form.

Simplify each side.

Subtract 36 from both sides.

Factor.

 62 = x(x - 5)

36 = x2 - 5x

0 = x2 - 5x - 36

0 = (x - 9)(x + 4)

 

Use the Zero Product Property.

Solve for x.

x - 9 = 0

x = 9

or

or

x + 4

x

= 0

= -4

 

log6 x is defined only when x is positive. Therefore, x = -4 is not a solution.

Thus, x = 9 is the solution of log6 x + log6(x - 5) = 2. We leave the check to you.

 

Example 2

Solve: ln 20x2 = 4.73 + ln 5x.

Step 1 Rewrite the equation with the logs on one side and the constant term on the other side.

Subtract ln 5x from both sides.

ln 20x2 = 4.73 + ln 5x
Step 2 Combine the logs into a single log.  
 

Use the Log of a Quotient Property.

  Simplify the argument of the log.

Write ln 4x as loge 4x.

 ln 4x = 4.73

 loge 4x = 4.73

Step 3 Convert the equation to exponential form and solve.

Convert to exponential form.

 e4.73 = 4x
  Divide both sides by 4.
  Find e4.73 and leave the result on the calculator display. Then, divide by 4 and round. 28.32 ≈ x
So, x 28.32. We leave the check to you.
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