Question
Part 1 of 2 HW Scores \( 68.13 \%, 436 \) of CA points potinter of 1 Solve 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 solut \[ 11^{x}=43 \] The solution set expressed in terms of logarithms is \( \square \) \}. (Use a comma to separate answers as needed. Simplify Vour answer. Use
Ask by Potter Parry. in Saudi Arabia
Mar 09,2025
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Answer
The solution set expressed in terms of logarithms is \( x = \frac{\ln(43)}{\ln(11)} \). Using a calculator, the decimal approximation is approximately \( x \approx 1.568 \).
Solution
We start with the given equation
\[
11^{x} = 43.
\]
Taking the natural logarithm of both sides gives
\[
\ln(11^x) = \ln(43).
\]
Using the power rule for logarithms, \(\ln(11^x) = x \ln(11)\), we have
\[
x \ln(11) = \ln(43).
\]
Solving for \(x\) by dividing both sides by \(\ln(11)\) yields
\[
x = \frac{\ln(43)}{\ln(11)}.
\]
Thus, the solution set expressed in terms of logarithms is
\[
\{x = \frac{\ln(43)}{\ln(11)}\}.
\]
Using a calculator for a decimal approximation:
\[
\ln(43) \approx 3.7612, \quad \ln(11) \approx 2.3979,
\]
\[
x \approx \frac{3.7612}{2.3979} \approx 1.568.
\]
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Simplify this solution Mind Expander
To solve the equation \( 11^{x} = 43 \), we can take the natural logarithm of both sides: \[ \ln(11^{x}) = \ln(43) \] Using the power rule of logarithms, we get: \[ x \cdot \ln(11) = \ln(43) \] Now, solving for \( x \) gives us: \[ x = \frac{\ln(43)}{\ln(11)} \] Now, let's plug this into a calculator to find the decimal approximation. You’ll get approximately \( x \approx 1.579 \). So the solution set expressed in terms of logarithms is \( \left\{ \frac{\ln(43)}{\ln(11)} \right\} \).
