Question
Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer.
log left parenthesis x plus 4 right parenthesis minus log 2 equals log left parenthesis 5 x plus 1 right parenthesis
Question content area bottom
Part 1
Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There are infinitely many solutions.
C.
There is no solution.
log left parenthesis x plus 4 right parenthesis minus log 2 equals log left parenthesis 5 x plus 1 right parenthesis
Question content area bottom
Part 1
Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .
(Simplify your answer. Use a comma to separate answers as needed.)
B.
There are infinitely many solutions.
C.
There is no solution.
Answers
GPT-4o mini
To solve the equation
\[
\log(x + 4) - \log(2) = \log(5x + 1),
\]
we can apply the properties of logarithms. Specifically, we can use the property that states \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\). This gives us
\[
\log\left(\frac{x + 4}{2}\right) = \log(5x + 1).
\]
Since both sides of the equation are logarithms, we can eliminate the logarithms by setting the arguments equal to each other, provided that the arguments are positive:
\[
\frac{x + 4}{2} = 5x + 1.
\]
Next, we will multiply both sides by 2 to eliminate the denominator:
\[
x + 4 = 10x + 2.
\]
Now, we can rearrange the equation to isolate \(x\):
\[
4 - 2 = 10x - x,
\]
which simplifies to:
\[
2 = 9x.
\]
Dividing both sides by 9 gives:
\[
x = \frac{2}{9}.
\]
Now, we need to check if this solution is in the domain of the original logarithmic expression. The arguments of the logarithms must be positive:
1. For \(\log(x + 4)\):
\[
x + 4 > 0 \implies \frac{2}{9} + 4 > 0 \text{ (which is true)}.
\]
2. For \(\log(2)\):
\(\log(2)\) is defined as 2 is a positive number.
3. For \(\log(5x + 1)\):
\[
5x + 1 > 0 \implies 5\left(\frac{2}{9}\right) + 1 = \frac{10}{9} + 1 = \frac{10}{9} + \frac{9}{9} = \frac{19}{9} > 0 \text{ (which is true)}.
\]
Since \(x = \frac{2}{9}\) is in the domain of the logarithmic functions, it is a valid solution.
Thus, the solution set is:
\[
\boxed{\left\{\frac{2}{9}\right\}}.
\]
\[
\log(x + 4) - \log(2) = \log(5x + 1),
\]
we can apply the properties of logarithms. Specifically, we can use the property that states \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\). This gives us
\[
\log\left(\frac{x + 4}{2}\right) = \log(5x + 1).
\]
Since both sides of the equation are logarithms, we can eliminate the logarithms by setting the arguments equal to each other, provided that the arguments are positive:
\[
\frac{x + 4}{2} = 5x + 1.
\]
Next, we will multiply both sides by 2 to eliminate the denominator:
\[
x + 4 = 10x + 2.
\]
Now, we can rearrange the equation to isolate \(x\):
\[
4 - 2 = 10x - x,
\]
which simplifies to:
\[
2 = 9x.
\]
Dividing both sides by 9 gives:
\[
x = \frac{2}{9}.
\]
Now, we need to check if this solution is in the domain of the original logarithmic expression. The arguments of the logarithms must be positive:
1. For \(\log(x + 4)\):
\[
x + 4 > 0 \implies \frac{2}{9} + 4 > 0 \text{ (which is true)}.
\]
2. For \(\log(2)\):
\(\log(2)\) is defined as 2 is a positive number.
3. For \(\log(5x + 1)\):
\[
5x + 1 > 0 \implies 5\left(\frac{2}{9}\right) + 1 = \frac{10}{9} + 1 = \frac{10}{9} + \frac{9}{9} = \frac{19}{9} > 0 \text{ (which is true)}.
\]
Since \(x = \frac{2}{9}\) is in the domain of the logarithmic functions, it is a valid solution.
Thus, the solution set is:
\[
\boxed{\left\{\frac{2}{9}\right\}}.
\]