Question
Use the laws of exponents to solve for x in the equation 414x=434
.(1 point)
Responses
16
16
12
Start Fraction 1 over 2 End Fraction
−12
negative Start Fraction 1 over 2 End Fraction
2
.(1 point)
Responses
16
16
12
Start Fraction 1 over 2 End Fraction
−12
negative Start Fraction 1 over 2 End Fraction
2
Answers
GPT-4o mini
To solve the equation \( 414^x = 434 \), we can start by taking the logarithm of both sides.
Using logarithms:
\[
\log(414^x) = \log(434)
\]
Using the power property of logarithms, we can bring down the exponent \( x \):
\[
x \cdot \log(414) = \log(434)
\]
Now, isolate \( x \):
\[
x = \frac{\log(434)}{\log(414)}
\]
Using a calculator to find the values of \( \log(434) \) and \( \log(414) \):
- \( \log(434) \approx 2.6387 \)
- \( \log(414) \approx 2.6180 \)
Now plug these values into the equation for \( x \):
\[
x \approx \frac{2.6387}{2.6180} \approx 1.0078
\]
Since none of the provided responses (16, 12, −12, \( \frac{1}{2} \), 2) match this answer, it seems that either the problem might involve some simplifications or the answers could apply in some other context related to the exponent itself.
If you are supposed to find \( x \) purely based on exponentiation properties without any additional calculations, it may not yield one of the provided answers.
For solving \( 414^x = 434 \) exactly, it doesn’t yield a simple integer or fraction based answer, unlike what the given set of answers implies.
Please check the original equation or the context for possible simplifications that might lead to one of the options, or recheck your available responses.
Using logarithms:
\[
\log(414^x) = \log(434)
\]
Using the power property of logarithms, we can bring down the exponent \( x \):
\[
x \cdot \log(414) = \log(434)
\]
Now, isolate \( x \):
\[
x = \frac{\log(434)}{\log(414)}
\]
Using a calculator to find the values of \( \log(434) \) and \( \log(414) \):
- \( \log(434) \approx 2.6387 \)
- \( \log(414) \approx 2.6180 \)
Now plug these values into the equation for \( x \):
\[
x \approx \frac{2.6387}{2.6180} \approx 1.0078
\]
Since none of the provided responses (16, 12, −12, \( \frac{1}{2} \), 2) match this answer, it seems that either the problem might involve some simplifications or the answers could apply in some other context related to the exponent itself.
If you are supposed to find \( x \) purely based on exponentiation properties without any additional calculations, it may not yield one of the provided answers.
For solving \( 414^x = 434 \) exactly, it doesn’t yield a simple integer or fraction based answer, unlike what the given set of answers implies.
Please check the original equation or the context for possible simplifications that might lead to one of the options, or recheck your available responses.