What is the solution of $\log (2 t+4)=\log (14-3 t)$?
Answer (1)
Set the arguments of the logarithms equal to each other: 2 t + 4 = 14 − 3 t .
Solve the equation for t : 5 t = 10 , which gives t = 2 .
Check if the solution is valid by plugging t = 2 into the original expressions: 0"> 2 ( 2 ) + 4 > 0 and 0"> 14 − 3 ( 2 ) > 0 .
Since the solution is valid, the final answer is 2 .
Explanation
Understanding the problem We are given the equation lo g ( 2 t + 4 ) = lo g ( 14 − 3 t ) . Our goal is to find the value of t that satisfies this equation.
Equating the arguments Since the logarithms of two expressions are equal, it means that the expressions themselves must be equal. Therefore, we can set the arguments of the logarithms equal to each other: 2 t + 4 = 14 − 3 t
Adding 3t to both sides Now, let's solve the equation for t . First, we add 3 t to both sides of the equation: 2 t + 3 t + 4 = 14 − 3 t + 3 t 5 t + 4 = 14
Subtracting 4 from both sides Next, we subtract 4 from both sides of the equation: 5 t + 4 − 4 = 14 − 4 5 t = 10
Dividing both sides by 5 Finally, we divide both sides by 5 to isolate t : 5 5 t = 5 10 t = 2
Checking the solution Now, we need to check if this solution is valid. The logarithm function is only defined for positive arguments. So, we need to make sure that 0"> 2 t + 4 > 0 and 0"> 14 − 3 t > 0 . Let's plug in t = 2 into both expressions:
For the first expression: 0"> 2 ( 2 ) + 4 = 4 + 4 = 8 > 0
For the second expression: 0"> 14 − 3 ( 2 ) = 14 − 6 = 8 > 0
Since both expressions are positive when t = 2 , the solution is valid.
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 in chemistry, and modeling population growth or decay in biology. For example, if you want to determine how long it will take for a population to double given a certain growth rate, you can use logarithmic equations to solve for the time variable.
