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# Solving Equations with Log Terms and Other Terms

Hereâ€™s how to solve an equation that contains more than one log and a constant term.

Procedure â€” 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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