Question Determine if the statement is correct: Since $3^{x}=20$ and $3^{x}=243$ are similar, they can be solved the same way.

Studdy Solution

STEP 1

Assumptions
1. We are comparing the methods to solve the equations $3^{x}=20$ and $3^{x}=243$ .
2. The methods considered are taking the logarithm of both sides and expressing each side as a power of the same base.

STEP 2

Let's first consider the equation $3^{x}=20$ . Since 20 is not a power of 3, we cannot directly express 20 as $3^{y}$ for some integer $y$ . Therefore, the method of expressing each side as a power of the same base is not applicable in this case.

STEP 3

To solve the equation $3^{x}=20$ , we can take the logarithm of both sides. This allows us to solve for $x$ even though 20 is not a power of 3.
$\log(3^{x}) = \log(20)$

STEP 4

Using the property of logarithms that $\log(a^{b}) = b \cdot \log(a)$ , we can rewrite the left side of the equation.
$x \cdot \log(3) = \log(20)$

STEP 5

Now, we can solve for $x$ by dividing both sides of the equation by $\log(3)$ .
$x = \frac{\log(20)}{\log(3)}$

STEP 6

Next, let's consider the equation $3^{x}=243$ . Since 243 is a power of 3 ( $243 = 3^5$ ), we can express both sides of the equation as a power of the same base.

STEP 7

We rewrite 243 as a power of 3.
$3^{x} = 3^5$

STEP 8

Since the bases are the same and the exponential function is one-to-one, we can equate the exponents.
$x = 5$

STEP 9

Now, we compare the methods used to solve both equations. For $3^{x}=20$ , we used logarithms, and for $3^{x}=243$ , we expressed each side as a power of the same base.

STEP 10

Based on the analysis in the previous steps, we can conclude that the statement "Because the equations $3^{x}=20$ and $3^{x}=243$ are similar, they can be solved using the same method" does not make sense because the methods used to solve each equation are different.

STEP 11

The correct answer is A. The statement does not make sense because the equation $3^{x}=20$ is solved by taking the logarithm on both sides, but the equation $3^{x}=243$ is solved by expressing each side as a power of the same base.

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Question Determine if the statement is correct: Since $3^{x}=20$ and $3^{x}=243$ are similar, they can be solved the same way.

Studdy Solution

STEP 1

Assumptions
1. We are comparing the methods to solve the equations $3^{x}=20$ and $3^{x}=243$ .
2. The methods considered are taking the logarithm of both sides and expressing each side as a power of the same base.

STEP 2

Let's first consider the equation $3^{x}=20$ . Since 20 is not a power of 3, we cannot directly express 20 as $3^{y}$ for some integer $y$ . Therefore, the method of expressing each side as a power of the same base is not applicable in this case.

STEP 3

To solve the equation $3^{x}=20$ , we can take the logarithm of both sides. This allows us to solve for $x$ even though 20 is not a power of 3.
$\log(3^{x}) = \log(20)$

STEP 4

Using the property of logarithms that $\log(a^{b}) = b \cdot \log(a)$ , we can rewrite the left side of the equation.
$x \cdot \log(3) = \log(20)$

STEP 5

Now, we can solve for $x$ by dividing both sides of the equation by $\log(3)$ .
$x = \frac{\log(20)}{\log(3)}$

STEP 6

Next, let's consider the equation $3^{x}=243$ . Since 243 is a power of 3 ( $243 = 3^5$ ), we can express both sides of the equation as a power of the same base.

STEP 7

We rewrite 243 as a power of 3.
$3^{x} = 3^5$

STEP 8

Since the bases are the same and the exponential function is one-to-one, we can equate the exponents.
$x = 5$

STEP 9

Now, we compare the methods used to solve both equations. For $3^{x}=20$ , we used logarithms, and for $3^{x}=243$ , we expressed each side as a power of the same base.

STEP 10

Based on the analysis in the previous steps, we can conclude that the statement "Because the equations $3^{x}=20$ and $3^{x}=243$ are similar, they can be solved using the same method" does not make sense because the methods used to solve each equation are different.

STEP 11

The correct answer is A. The statement does not make sense because the equation $3^{x}=20$ is solved by taking the logarithm on both sides, but the equation $3^{x}=243$ is solved by expressing each side as a power of the same base.

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Question Determine if the statement is correct: Since 3x=203^{x}=203x=20 and 3x=2433^{x}=2433x=243 are similar, they can be solved the same way.

Studdy Solution

STEP 1

Assumptions1. We are comparing the methods to solve the equations 3x=203^{x}=203x=20 and 3x=2433^{x}=2433x=243.2. The methods considered are taking the logarithm of both sides and expressing each side as a power of the same base.

STEP 2

Let's first consider the equation 3x=203^{x}=203x=20. Since 20 is not a power of 3, we cannot directly express 20 as 3y3^{y}3y for some integer yyy. Therefore, the method of expressing each side as a power of the same base is not applicable in this case.

STEP 3

To solve the equation 3x=203^{x}=203x=20, we can take the logarithm of both sides. This allows us to solve for xxx even though 20 is not a power of 3.log⁡(3x)=log⁡(20)\log(3^{x}) = \log(20)log(3x)=log(20)

STEP 4

Using the property of logarithms that log⁡(ab)=b⋅log⁡(a)\log(a^{b}) = b \cdot \log(a)log(ab)=b⋅log(a), we can rewrite the left side of the equation.x⋅log⁡(3)=log⁡(20)x \cdot \log(3) = \log(20)x⋅log(3)=log(20)

STEP 5

Now, we can solve for xxx by dividing both sides of the equation by log⁡(3)\log(3)log(3).x=log⁡(20)log⁡(3)x = \frac{\log(20)}{\log(3)}x=log(3)log(20)​

STEP 6

Next, let's consider the equation 3x=2433^{x}=2433x=243. Since 243 is a power of 3 (243=35243 = 3^5243=35), we can express both sides of the equation as a power of the same base.

STEP 7

We rewrite 243 as a power of 3.3x=353^{x} = 3^53x=35

STEP 8

Since the bases are the same and the exponential function is one-to-one, we can equate the exponents.x=5x = 5x=5

STEP 9

Now, we compare the methods used to solve both equations. For 3x=203^{x}=203x=20, we used logarithms, and for 3x=2433^{x}=2433x=243, we expressed each side as a power of the same base.

STEP 10

Based on the analysis in the previous steps, we can conclude that the statement "Because the equations 3x=203^{x}=203x=20 and 3x=2433^{x}=2433x=243 are similar, they can be solved using the same method" does not make sense because the methods used to solve each equation are different.

STEP 11

The correct answer is A. The statement does not make sense because the equation 3x=203^{x}=203x=20 is solved by taking the logarithm on both sides, but the equation 3x=2433^{x}=2433x=243 is solved by expressing each side as a power of the same base.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Determine if the statement is correct: Since $3^{x}=20$ and $3^{x}=243$ are similar, they can be solved the same way.

Assumptions
1. We are comparing the methods to solve the equations $3^{x}=20$ and $3^{x}=243$ .
2. The methods considered are taking the logarithm of both sides and expressing each side as a power of the same base.

Let's first consider the equation $3^{x}=20$ . Since 20 is not a power of 3, we cannot directly express 20 as $3^{y}$ for some integer $y$ . Therefore, the method of expressing each side as a power of the same base is not applicable in this case.

To solve the equation $3^{x}=20$ , we can take the logarithm of both sides. This allows us to solve for $x$ even though 20 is not a power of 3.
$\log(3^{x}) = \log(20)$

Using the property of logarithms that $\log(a^{b}) = b \cdot \log(a)$ , we can rewrite the left side of the equation.
$x \cdot \log(3) = \log(20)$

Now, we can solve for $x$ by dividing both sides of the equation by $\log(3)$ .
$x = \frac{\log(20)}{\log(3)}$

Next, let's consider the equation $3^{x}=243$ . Since 243 is a power of 3 ( $243 = 3^5$ ), we can express both sides of the equation as a power of the same base.

We rewrite 243 as a power of 3.
$3^{x} = 3^5$

Since the bases are the same and the exponential function is one-to-one, we can equate the exponents.
$x = 5$

Now, we compare the methods used to solve both equations. For $3^{x}=20$ , we used logarithms, and for $3^{x}=243$ , we expressed each side as a power of the same base.

Based on the analysis in the previous steps, we can conclude that the statement "Because the equations $3^{x}=20$ and $3^{x}=243$ are similar, they can be solved using the same method" does not make sense because the methods used to solve each equation are different.

The correct answer is A. The statement does not make sense because the equation $3^{x}=20$ is solved by taking the logarithm on both sides, but the equation $3^{x}=243$ is solved by expressing each side as a power of the same base.