Math / Algebra

QuestionSolve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. $11^{x}=67$
The solution set expressed in terms of logarithms is $\square$ \}. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use In for natural logarithm and log for common logarithm.)

Studdy Solution

STEP 1

1. The equation $11^x = 67$ is an exponential equation.
2. We will use logarithms to solve for $x$ .
3. We can use either natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ).

STEP 2

1. Apply logarithms to both sides of the equation.
2. Use logarithmic properties to solve for $x$ .
3. Express the solution in terms of logarithms.
4. Calculate a decimal approximation using a calculator.

STEP 3

Apply logarithms to both sides of the equation. You can choose either natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ). Here, we'll use common logarithms:
$\log(11^x) = \log(67)$

STEP 4

Use the power rule of logarithms, which states that $\log(a^b) = b \cdot \log(a)$ , to bring the exponent $x$ down:
$x \cdot \log(11) = \log(67)$

STEP 5

Solve for $x$ by dividing both sides by $\log(11)$ :
$x = \frac{\log(67)}{\log(11)}$
This is the solution set expressed in terms of common logarithms.

STEP 6

Use a calculator to find a decimal approximation for $x$ . Calculate:
$x \approx \frac{\log(67)}{\log(11)}$
Using a calculator, we find:
$x \approx \frac{1.8261}{1.0414} \approx 1.753$
The solution set expressed in terms of logarithms is:
$\left\{ \frac{\log(67)}{\log(11)} \right\}$
The decimal approximation for the solution is approximately $x \approx 1.753$ .

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Studdy Solution

STEP 1

1. The equation $11^x = 67$ is an exponential equation.
2. We will use logarithms to solve for $x$ .
3. We can use either natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ).

STEP 2

1. Apply logarithms to both sides of the equation.
2. Use logarithmic properties to solve for $x$ .
3. Express the solution in terms of logarithms.
4. Calculate a decimal approximation using a calculator.

STEP 3

Apply logarithms to both sides of the equation. You can choose either natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ). Here, we'll use common logarithms:
$\log(11^x) = \log(67)$

STEP 4

Use the power rule of logarithms, which states that $\log(a^b) = b \cdot \log(a)$ , to bring the exponent $x$ down:
$x \cdot \log(11) = \log(67)$

STEP 5

Solve for $x$ by dividing both sides by $\log(11)$ :
$x = \frac{\log(67)}{\log(11)}$
This is the solution set expressed in terms of common logarithms.

STEP 6

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Studdy Solution

STEP 1

1. The equation 11x=67 11^x = 67 11x=67 is an exponential equation.2. We will use logarithms to solve for x x x.3. We can use either natural logarithms (ln⁡\lnln) or common logarithms (log⁡\loglog).

STEP 2

1. Apply logarithms to both sides of the equation.2. Use logarithmic properties to solve for x x x.3. Express the solution in terms of logarithms.4. Calculate a decimal approximation using a calculator.

STEP 3

Apply logarithms to both sides of the equation. You can choose either natural logarithms (ln⁡\lnln) or common logarithms (log⁡\loglog). Here, we'll use common logarithms:log⁡(11x)=log⁡(67) \log(11^x) = \log(67) log(11x)=log(67)

STEP 4

Use the power rule of logarithms, which states that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b \cdot \log(a)log(ab)=b⋅log(a), to bring the exponent xxx down:x⋅log⁡(11)=log⁡(67) x \cdot \log(11) = \log(67) x⋅log(11)=log(67)

STEP 5

Solve for x x x by dividing both sides by log⁡(11)\log(11)log(11):x=log⁡(67)log⁡(11) x = \frac{\log(67)}{\log(11)} x=log(11)log(67)​This is the solution set expressed in terms of common logarithms.

STEP 6

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1. The equation $11^x = 67$ is an exponential equation.
2. We will use logarithms to solve for $x$ .
3. We can use either natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ).

1. Apply logarithms to both sides of the equation.
2. Use logarithmic properties to solve for $x$ .
3. Express the solution in terms of logarithms.
4. Calculate a decimal approximation using a calculator.

Apply logarithms to both sides of the equation. You can choose either natural logarithms ( $\ln$ ) or common logarithms ( $\log$ ). Here, we'll use common logarithms:
$\log(11^x) = \log(67)$

Use the power rule of logarithms, which states that $\log(a^b) = b \cdot \log(a)$ , to bring the exponent $x$ down:
$x \cdot \log(11) = \log(67)$

Solve for $x$ by dividing both sides by $\log(11)$ :
$x = \frac{\log(67)}{\log(11)}$
This is the solution set expressed in terms of common logarithms.