Question

question 7 of 10
what is the solution to the equation below? round your answer to two decimal places.
7 9^x = 55.68

a. x = 0.94
b. x = 2.07
c. x = 1.45
d. x = 1.06

Explanation:

Step1: Isolate the exponential term

We have the equation \( 9^x = 55.68 \) (assuming the "7" is a typo or formatting error, as it doesn't make sense in the equation as written; likely the equation is \( 9^x = 55.68 \)). To solve for \( x \), we take the logarithm of both sides. Using the natural logarithm (ln) for convenience:
\( \ln(9^x) = \ln(55.68) \)

Step2: Apply logarithm power rule

The power rule of logarithms states that \( \ln(a^b) = b\ln(a) \). Applying this to the left side:
\( x\ln(9) = \ln(55.68) \)

Step3: Solve for \( x \)

Divide both sides by \( \ln(9) \):
\( x = \frac{\ln(55.68)}{\ln(9)} \)
Calculate the values: \( \ln(55.68) \approx 4.020 \), \( \ln(9) \approx 2.197 \)
\( x \approx \frac{4.020}{2.197} \approx 1.83 \) Wait, this doesn't match the options. Wait, maybe the original equation was \( 7 \times 9^x = 55.68 \). Let's correct that. If the equation is \( 7 \times 9^x = 55.68 \), then:

Step1 (corrected): Isolate the exponential term

Divide both sides by 7: \( 9^x = \frac{55.68}{7} = 7.954 \)

Step2 (corrected): Take logarithm of both sides

\( \ln(9^x) = \ln(7.954) \)

Step3 (corrected): Apply power rule

\( x\ln(9) = \ln(7.954) \)

Step4 (corrected): Solve for \( x \)

\( x = \frac{\ln(7.954)}{\ln(9)} \)
Calculate: \( \ln(7.954) \approx 2.075 \), \( \ln(9) \approx 2.197 \)
\( x \approx \frac{2.075}{2.197} \approx 0.94 \)
Which matches option A.

Answer:

A. \( x = 0.94 \)