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Find an approximate irrational solution to [tex]$3^x = 2$[/tex].

Select the correct choice below and, if necessary, fill in the answer box to complete your choice:

A. The solution set is \{ [tex]$\square$[/tex] \}. (Round to four decimal places.)

B. The solution is the empty set.

Sagot :

GinnyAnsweridn
To find an approximate solution to the equation [tex]\(3^x = 2\)[/tex], we can follow these steps:

1. Take the natural logarithm of both sides:

ln([tex]\(3^x\)[/tex]) = ln(2)

2. Use the properties of logarithms (specifically, the power rule [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]):

[tex]\(x \cdot \ln(3) = \ln(2)\)[/tex]

3. Solve for [tex]\(x\)[/tex]:

[tex]\( x = \frac{\ln(2)}{\ln(3)} \)[/tex]

Using numerical values from logarithms:

- [tex]\(\ln(2) \approx 0.6931\)[/tex]
- [tex]\(\ln(3) \approx 1.0986\)[/tex]

So,

[tex]\[ x \approx \frac{0.6931}{1.0986} \approx 0.6309297535714574 \][/tex]

When rounded to four decimal places, the solution is:

[tex]\[ x \approx 0.6309 \][/tex]

Thus, the correct choice is:

A. The solution set is [tex]\( \{ 0.6309 \} \)[/tex].
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