Solve for the values of $k$ for which $\log (3 k+3)+\log k-\log 36=0$.
Answer (2)
The solution to the equation lo g ( 3 k + 3 ) + lo g k − lo g 36 = 0 is found by combining and simplifying the logarithms, ultimately leading to a quadratic equation. The valid result for k that satisfies the conditions of the logarithm is 3 . Therefore, the final answer is 3 .
;
Combine the logarithms using the properties lo g a + lo g b = lo g ( ab ) and lo g a − lo g b = lo g ( b a ) .
Remove the logarithm by exponentiating both sides.
Simplify the equation to a quadratic equation k 2 + k − 12 = 0 .
Solve the quadratic equation by factoring to find the valid solution k = 3 .
3
Explanation
Understanding the Problem and Constraints We are given the equation lo g ( 3 k + 3 ) + lo g k − lo g 36 = 0 . We need to solve for the values of k . The domain of the logarithm function requires that 0"> 3 k + 3 > 0 and 0"> k > 0 , which means -1"> k > − 1 and 0"> k > 0 . Thus, we must have 0"> k > 0 .
Combining Logarithms Using the logarithm property lo g a + lo g b = lo g ( ab ) , we can combine the first two terms: lo g (( 3 k + 3 ) k ) − lo g 36 = 0
Further Combining Logarithms Using the logarithm property lo g a − lo g b = lo g ( b a ) , we can combine the terms: lo g ( 36 ( 3 k + 3 ) k ) = 0
Removing the Logarithm To remove the logarithm, we exponentiate both sides with base 10: 36 ( 3 k + 3 ) k = 1 0 0 = 1
Simplifying the Equation Now, we simplify the equation: ( 3 k + 3 ) k = 36 Expanding the equation gives: 3 k 2 + 3 k = 36
Forming a Quadratic Equation Rearranging the equation into a quadratic equation: 3 k 2 + 3 k − 36 = 0 Dividing the equation by 3: k 2 + k − 12 = 0
Solving the Quadratic Equation We can solve the quadratic equation by factoring: k 2 + k − 12 = ( k + 4 ) ( k − 3 ) = 0 Thus, the solutions are k = − 4 and k = 3 .
Checking for Valid Solutions However, we must check if the solutions satisfy the condition 0"> k > 0 . Since k must be greater than 0, k = − 4 is not a valid solution. Therefore, the only valid solution is k = 3 .
Final Answer Thus, the value of k for which lo g ( 3 k + 3 ) + lo g k − lo g 36 = 0 is k = 3 .
Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and modeling population growth or decay. In finance, they are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding how to solve logarithmic equations is essential for making informed decisions and predictions in these areas.
