Solve the equation: $\log _3(x)+\log _3(x+8)=2$
The solution is $x=$ $\square$
Answer (2)
The solution to the equation lo g 3 ( x ) + lo g 3 ( x + 8 ) = 2 is x = 1 . After combining logarithms and converting to exponential form, we formed a quadratic equation and found that the only valid solution is x = 1 . We confirmed that x = − 9 is extraneous as it does not satisfy the logarithmic conditions.
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Combine the logarithms using the property lo g a ( b ) + lo g a ( c ) = lo g a ( b c ) .
Convert the logarithmic equation to an exponential equation.
Solve the resulting quadratic equation by factoring.
Check for extraneous solutions to find the valid solution: 1 .
Explanation
Problem Analysis We are given the equation lo g 3 ( x ) + lo g 3 ( x + 8 ) = 2 and we need to solve for x .
Combining Logarithms First, we use the logarithm property lo g a ( b ) + lo g a ( c ) = lo g a ( b c ) to combine the logarithms on the left side of the equation: lo g 3 ( x ( x + 8 )) = 2
Converting to Exponential Form Next, we convert the logarithmic equation to an exponential equation: x ( x + 8 ) = 3 2
Simplifying the Equation Now, we simplify the equation: x 2 + 8 x = 9
Rearranging to Quadratic Form Rearrange the equation into a quadratic equation: x 2 + 8 x − 9 = 0
Factoring the Quadratic We factor the quadratic equation: ( x + 9 ) ( x − 1 ) = 0
Solving for x Now we solve for x : x = − 9 or x = 1
Checking for Extraneous Solutions We need to check for extraneous solutions by plugging the values of x back into the original equation. Since the logarithm of a negative number is undefined, x = − 9 is an extraneous solution because lo g 3 ( − 9 ) is undefined. Therefore, the only valid solution is x = 1 .
Final Answer Thus, the solution to the equation is x = 1 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. Understanding how to solve logarithmic equations is crucial for making accurate predictions and interpretations in these areas. For example, if we know the intensity of an earthquake, we can use a logarithmic equation to determine its magnitude.
