Math / Algebra

Question( Lab Homework Question 13, 4.4-20 Points: of of 1
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. $\log _{3} x=4$ A. $\{1.262\}$ B. (64) C. (12) D. (81)

Studdy Solution

STEP 1

1. The equation given is $\log_{3} x = 4$ , which involves the logarithmic function with base 3.
2. To solve for $x$ , we need to convert the logarithmic equation to its exponential form.
3. Logarithmic equations may have specific domain restrictions; for $\log_{3} x$ , $x$ must be greater than 0.

STEP 2

1. Convert the logarithmic equation to its equivalent exponential form.
2. Solve the resulting exponential equation for $x$ .
3. Verify the solution to ensure it is within the domain of the original logarithmic function.

STEP 3

Convert the logarithmic equation $\log_{3} x = 4$ to its equivalent exponential form. Recall that $\log_{b} a = c$ is equivalent to $a = b^c$ .
$x = 3^4$

STEP 4

Calculate $3^4$ to find the value of $x$ .
$3^4 = 3 \times 3 \times 3 \times 3 = 81$

STEP 5

Verify that the solution $x = 81$ is within the domain of the original logarithmic function $\log_{3} x$ . Since $81 > 0$ , it is within the domain.
The solution to the equation $\log_{3} x = 4$ is:
$x = 81$
The correct answer is: D. (81)

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Question( Lab Homework Question 13, 4.4-20 Points: of of 1
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. $\log _{3} x=4$ A. $\{1.262\}$ B. (64) C. (12) D. (81)

Studdy Solution

STEP 1

1. The equation given is $\log_{3} x = 4$ , which involves the logarithmic function with base 3.
2. To solve for $x$ , we need to convert the logarithmic equation to its exponential form.
3. Logarithmic equations may have specific domain restrictions; for $\log_{3} x$ , $x$ must be greater than 0.

STEP 2

1. Convert the logarithmic equation to its equivalent exponential form.
2. Solve the resulting exponential equation for $x$ .
3. Verify the solution to ensure it is within the domain of the original logarithmic function.

STEP 3

Convert the logarithmic equation $\log_{3} x = 4$ to its equivalent exponential form. Recall that $\log_{b} a = c$ is equivalent to $a = b^c$ .
$x = 3^4$

STEP 4

Calculate $3^4$ to find the value of $x$ .
$3^4 = 3 \times 3 \times 3 \times 3 = 81$

STEP 5

Verify that the solution $x = 81$ is within the domain of the original logarithmic function $\log_{3} x$ . Since $81 > 0$ , it is within the domain.
The solution to the equation $\log_{3} x = 4$ is:
$x = 81$
The correct answer is: D. (81)

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Question( Lab Homework Question 13, 4.4-20 Points: of of 1Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. log⁡3x=4\log _{3} x=4log3​x=4 A. {1.262}\{1.262\}{1.262} B. (64) C. (12) D. (81)

Studdy Solution

STEP 1

STEP 2

1. Convert the logarithmic equation to its equivalent exponential form.2. Solve the resulting exponential equation for xxx.3. Verify the solution to ensure it is within the domain of the original logarithmic function.

STEP 3

Convert the logarithmic equation log⁡3x=4\log_{3} x = 4log3​x=4 to its equivalent exponential form. Recall that log⁡ba=c\log_{b} a = clogb​a=c is equivalent to a=bca = b^ca=bc.x=34 x = 3^4 x=34

STEP 4

Calculate 343^434 to find the value of xxx.34=3×3×3×3=81 3^4 = 3 \times 3 \times 3 \times 3 = 81 34=3×3×3×3=81

STEP 5

Verify that the solution x=81x = 81x=81 is within the domain of the original logarithmic function log⁡3x\log_{3} xlog3​x. Since 81>081 > 081>0, it is within the domain. The solution to the equation log⁡3x=4\log_{3} x = 4log3​x=4 is:x=81 x = 81 x=81The correct answer is: D. (81)

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Question( Lab Homework Question 13, 4.4-20 Points: of of 1
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. $\log _{3} x=4$ A. $\{1.262\}$ B. (64) C. (12) D. (81)

1. Convert the logarithmic equation to its equivalent exponential form.
2. Solve the resulting exponential equation for $x$ .
3. Verify the solution to ensure it is within the domain of the original logarithmic function.

Convert the logarithmic equation $\log_{3} x = 4$ to its equivalent exponential form. Recall that $\log_{b} a = c$ is equivalent to $a = b^c$ .
$x = 3^4$

Calculate $3^4$ to find the value of $x$ .
$3^4 = 3 \times 3 \times 3 \times 3 = 81$

Verify that the solution $x = 81$ is within the domain of the original logarithmic function $\log_{3} x$ . Since $81 > 0$ , it is within the domain.
The solution to the equation $\log_{3} x = 4$ is:
$x = 81$
The correct answer is: D. (81)