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Solving Equations with Log Terms on Each Side

When solving equations with two or more logs it is often necessary to use the properties of logarithms.

Log of a Product Property:

Log of a Quotient Property:

Log of a Power Property:

logb u · v = logb u + logb v

logb = logb u - logb v

logb un = n · logbu

 

Here is another useful property. It says that if the logs of two quantities are equal, then the quantities are equal.

 

Property — Logarithmic Equality

If logb x = logb y, then x = y.

Here, x > 0, y > 0, b > 0 and b 1.

 

In the following examples, we will first use the properties of logarithms to rewrite each equation so there is one log on each side. Then, we will use the Principal of Logarithmic Equality.

 

Example 1

Solve: log 4 + log x = log 14

Solution

Use the Log of a Product Property.

Use the Principal of Logarithmic Equality.

Divide both sides by 4.

log 4 + log x

log 4x

4x

x

= log 14

= log 14

= 14

= 3.5

 So, the solution is x = 3.5.

 

Note:

To check the solution of log 4 + log x = log 14, substitute 3.5 for x and simplify.

 

Is

Is

Is

 log 4 + log x

log 4 + log 3.5

0.602 + 0.544

1.146

= log 14

= log 14 ?

= 1.146 ?

= 1.146 = Yes

Because we must round the values of the logs, in some cases the two sides of the equation will not simplify to exactly the same result. However, the results should be very close.

 

Example 2

Solve: log (x + 4) - log 6 = log (x - 11).

Solution log (x + 4) - log 6 = log (x - 11)
Use the Log of a Quotient Property. = log (x - 11)
Use the Principal of Logarithmic Equality. = x - 11
Multiply both sides by 6.

Subtract x from both sides.

Add 66 to both sides.

Divide both sides by 5.

 x + 4

4

70

14

= 6x - 66

= 5x - 66

= 5x

= x

So, the solution is x = 14. We leave the check to you.
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