Solving equations Solve each equation.

ln 3x ...

Question

Solving equations Solve each equation.

ln 3x + ln (x + 2) = 0

Accepted Answer

Welcome back, everyone. Determine the value of Z that satisfies LN 5 Z plus LN of Z + 4 equals 0. We're given 4 answer choices. A says Z equals 2 + +25 divided by 5. B 2 minus 25 divided by 5. C 2 minus 5 divided by 5, and D-2 + 25 divided by 5. So we want to solve an equation LN of 5 Z. Plus Ln of Z + 4 equals 0 and before we solve an equation in terms of logs, we want to evaluate the domain of each function. We know that natural logs cannot have a negative number inside of them, right? So first of all, 5 Z must be positive and Z + 4 must be positive as well. And we want to evaluate the overall domain. Solving the first inequality, we see that Z must be positive. And for the second equation, if we subtract 4 from both sides, the inequality becomes Z greater than -4. And overall, Z must be positive to satisfy each of them. Now, let's use one of the properties of locks. Since we have the addition of two locks with the same basis, we can rewrite this as a single log of the product between 5 Z and Z + 4. This must be equal to 0. Now if we use the inverse of log, which is an exponential function, e to the power of 0. Must be equal to 5 Z multiplied by Z + 4, or we can simply recall that LN of 1 is equal to 0, right? And each the power of 0 is 1. So 5C multiplied by Z + 4 must be equal to 1, and now we want to distribute. The terms, so we get 5 z squared. Plus 20 Z and we can subtract one from both sides to end up with zero because we want to arrive at a quadratic equation where A is 5. B is 20 and C is -1. And using the quadratic formula, we get two solutions Z1 and Z2. Would be equal to negative b plus or minus square root of b quad minus 4 AC. Divided by 28. Let's substitute the values we get -20 plus or minus square root of. B2, which is 20 2 minus 4 multiplied by 5 multiplied by -1. Divided by 2 multiplied by 5. We get Z1 equals. Now, -20 divided by 10 gives us -2. We first will take the positive value of the square root and we end up with square root of 400 plus 20, right? And that's 440. So square root of 440. Divided by 10. Now let's simplify. First of all, we can notice that 440 is divisible by 4, right? So we can isolate squared of 105 and put 2 in front of the radical because we're taking square of 4, which gives us 2. So Z1 can be written as -2. Plus Square root of 105. And because we have 2 in front, 2 divided by 10 is the same as 1 divided by 5, right? So we can simplify. And we will have the 2nd solution with a negative sign, but Z must be positive, right? So we can only get one solution with a positive sign. And this means that we have our final answer. So if we look at the answer choices, we can conclude that the correct answer to this problem corresponds to the answer choice D. Thank you for watching.

Solving equations Solve each equation.

ln 3x + ln (x + 2) = 0

Key Concepts

Properties of Logarithms

Exponential Functions

Domain of Logarithmic Functions

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