Math / Algebra

QuestionSolve 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.
$\log(x + 10) - \log 2 = \log(4x + 3)$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes the equation $\log(x + 10) - \log 2 = \log(4x + 3)$ true, while making sure the inputs to the log function are positive! Watch out! Remember that the logarithm of a negative number or zero is undefined, so we must check our solutions!

STEP 2

1. Combine the logs
2. Remove the logs
3. Solve for $x$
4. Check the solution

STEP 3

We have a difference of logs on the left side.
Remember the log rule: $\log a - \log b = \log(\frac{a}{b})$ .
So, $\log(x + 10) - \log 2$ becomes $\log(\frac{x + 10}{2})$ .
Our equation now looks like this: $\log(\frac{x + 10}{2}) = \log(4x + 3)$ .
This is great progress!

STEP 4

Since we have $\log(\text{something}) = \log(\text{something else})$ , then those "somethings" must be equal!
That means $\frac{x + 10}{2} = 4x + 3$ .
One step closer to finding $x$ !

STEP 5

To get rid of the fraction, let's multiply both sides of our equation by 2: $2 \cdot \frac{x + 10}{2} = 2 \cdot (4x + 3)$ $x + 10 = 8x + 6$

STEP 6

Now, subtract $x$ from both sides to get all the $x$ terms together: $x + 10 - x = 8x + 6 - x$ $10 = 7x + 6$

STEP 7

Let's subtract 6 from both sides to isolate the term with $x$ : $10 - 6 = 7x + 6 - 6$ $4 = 7x$

STEP 8

Finally, divide both sides by 7 to find $x$ : $\frac{4}{7} = \frac{7x}{7}$ $x = \frac{4}{7}$

STEP 9

We found $x = \frac{4}{7}$ .
Let's make sure this value doesn't make the arguments of the original logarithms negative or zero. For $\log(x + 10)$ , we have $\frac{4}{7} + 10 = \frac{4 + 70}{7} = \frac{74}{7}$ , which is positive! For $\log(4x + 3)$ , we have $4 \cdot \frac{4}{7} + 3 = \frac{16}{7} + 3 = \frac{16 + 21}{7} = \frac{37}{7}$ , which is also positive! So, our solution is valid!

STEP 10

The solution to the equation is $x = \frac{4}{7}$ .

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QuestionSolve 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.
$\log(x + 10) - \log 2 = \log(4x + 3)$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes the equation $\log(x + 10) - \log 2 = \log(4x + 3)$ true, while making sure the inputs to the log function are positive! Watch out! Remember that the logarithm of a negative number or zero is undefined, so we must check our solutions!

STEP 2

1. Combine the logs
2. Remove the logs
3. Solve for $x$
4. Check the solution

STEP 3

We have a difference of logs on the left side.
Remember the log rule: $\log a - \log b = \log(\frac{a}{b})$ .
So, $\log(x + 10) - \log 2$ becomes $\log(\frac{x + 10}{2})$ .
Our equation now looks like this: $\log(\frac{x + 10}{2}) = \log(4x + 3)$ .
This is great progress!

STEP 4

Since we have $\log(\text{something}) = \log(\text{something else})$ , then those "somethings" must be equal!
That means $\frac{x + 10}{2} = 4x + 3$ .
One step closer to finding $x$ !

STEP 5

To get rid of the fraction, let's multiply both sides of our equation by 2: $2 \cdot \frac{x + 10}{2} = 2 \cdot (4x + 3)$ $x + 10 = 8x + 6$

STEP 6

Now, subtract $x$ from both sides to get all the $x$ terms together: $x + 10 - x = 8x + 6 - x$ $10 = 7x + 6$

STEP 7

Let's subtract 6 from both sides to isolate the term with $x$ : $10 - 6 = 7x + 6 - 6$ $4 = 7x$

STEP 8

Finally, divide both sides by 7 to find $x$ : $\frac{4}{7} = \frac{7x}{7}$ $x = \frac{4}{7}$

STEP 9

STEP 10

The solution to the equation is $x = \frac{4}{7}$ .

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QuestionSolve 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⁡(x+10)−log⁡2=log⁡(4x+3)\log(x + 10) - \log 2 = \log(4x + 3)log(x+10)−log2=log(4x+3)

Studdy Solution

STEP 1

STEP 2

1. Combine the logs2. Remove the logs3. Solve for xxx4. Check the solution

STEP 3

STEP 4

Since we have log⁡(something)=log⁡(something else)\log(\text{something}) = \log(\text{something else})log(something)=log(something else), then those "somethings" must be equal!That means x+102=4x+3\frac{x + 10}{2} = 4x + 32x+10​=4x+3.One step closer to finding xxx!

STEP 5

To get rid of the fraction, let's multiply both sides of our equation by **2**: 2⋅x+102=2⋅(4x+3) 2 \cdot \frac{x + 10}{2} = 2 \cdot (4x + 3) 2⋅2x+10​=2⋅(4x+3) x+10=8x+6 x + 10 = 8x + 6 x+10=8x+6

STEP 6

Now, subtract xxx from both sides to get all the xxx terms together: x+10−x=8x+6−x x + 10 - x = 8x + 6 - x x+10−x=8x+6−x 10=7x+6 10 = 7x + 6 10=7x+6

STEP 7

Let's subtract 6 from both sides to isolate the term with xxx: 10−6=7x+6−6 10 - 6 = 7x + 6 - 6 10−6=7x+6−6 4=7x 4 = 7x 4=7x

STEP 8

Finally, divide both sides by **7** to find xxx: 47=7x7 \frac{4}{7} = \frac{7x}{7} 74​=77x​ x=47 x = \frac{4}{7} x=74​

STEP 9

STEP 10

The solution to the equation is x=47x = \frac{4}{7}x=74​.

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QuestionSolve 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.
$\log(x + 10) - \log 2 = \log(4x + 3)$

1. Combine the logs
2. Remove the logs
3. Solve for $x$
4. Check the solution

Since we have $\log(\text{something}) = \log(\text{something else})$ , then those "somethings" must be equal!
That means $\frac{x + 10}{2} = 4x + 3$ .
One step closer to finding $x$ !

To get rid of the fraction, let's multiply both sides of our equation by 2: $2 \cdot \frac{x + 10}{2} = 2 \cdot (4x + 3)$ $x + 10 = 8x + 6$

Now, subtract $x$ from both sides to get all the $x$ terms together: $x + 10 - x = 8x + 6 - x$ $10 = 7x + 6$

Let's subtract 6 from both sides to isolate the term with $x$ : $10 - 6 = 7x + 6 - 6$ $4 = 7x$

Finally, divide both sides by 7 to find $x$ : $\frac{4}{7} = \frac{7x}{7}$ $x = \frac{4}{7}$

The solution to the equation is $x = \frac{4}{7}$ .