Certainly! Let's solve the problem step-by-step:
The problem requires finding the power to which 6 must be raised in order to get the number 32768. In mathematical notation, this can be presented as:
[tex]\[6^x = 32768\][/tex]
We need to solve for [tex]\(x\)[/tex]. Here are the detailed steps:
1. Understand the Problem:
We need to determine what power [tex]\(x\)[/tex] will make [tex]\(6\)[/tex] equal to [tex]\(32768\)[/tex] when raised to [tex]\(x\)[/tex]. In other words, we are solving for [tex]\(x\)[/tex] in the equation [tex]\(6^x = 32768\)[/tex].
2. Logarithmic Transformation:
To solve for [tex]\(x\)[/tex], we can take the logarithm on both sides. We can use the logarithm base [tex]\(6\)[/tex], but to keep things simple and conventional, let's use the natural logarithm (base [tex]\(e\)[/tex]) or the common logarithm (base [tex]\(10\)[/tex]).
We'll use the natural logarithm for this example:
[tex]\[
\ln(6^x) = \ln(32768)
\][/tex]
3. Apply the Power Rule of Logarithms:
The power rule of logarithms states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Applying this rule, we get:
[tex]\[
x \cdot \ln(6) = \ln(32768)
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], divide both sides by [tex]\(\ln(6)\)[/tex]:
[tex]\[
x = \frac{\ln(32768)}{\ln(6)}
\][/tex]
5. Evaluate the Expression:
Here, [tex]\(\ln(32768)\)[/tex] and [tex]\(\ln(6)\)[/tex] are constant values. When evaluated, these values will provide a numerical solution for [tex]\(x\)[/tex]:
[tex]\[
x = 5.8027921085181235
\][/tex]
Therefore, the power to which 6 must be raised to obtain 32768 is approximately [tex]\(5.8027921085181235\)[/tex].
This detailed step-by-step solution shows that [tex]\(6^{5.8027921085181235} \approx 32768\)[/tex].
