To solve the logarithmic equation [tex]\(\log_{216}(6) = \frac{1}{3}\)[/tex], we need to convert it into its equivalent exponential form.
Recall that a logarithmic equation [tex]\(\log_b(a) = c\)[/tex] is equivalent to an exponential equation [tex]\(b^c = a\)[/tex], where:
- [tex]\(b\)[/tex] is the base of the logarithm,
- [tex]\(a\)[/tex] is the argument (the value you're taking the log of), and
- [tex]\(c\)[/tex] is the result of the logarithm.
Given the logarithmic equation [tex]\(\log_{216}(6) = \frac{1}{3}\)[/tex]:
- The base ([tex]\(b\)[/tex]) is 216,
- The argument ([tex]\(a\)[/tex]) is 6,
- The result ([tex]\(c\)[/tex]) is [tex]\(\frac{1}{3}\)[/tex].
We can rewrite this in exponential form as:
[tex]\[ 216^{\frac{1}{3}} = 6 \][/tex]
This shows that raising 216 to the power of [tex]\(\frac{1}{3}\)[/tex] gives us 6.
