To solve the logarithmic equation [tex]\(\log_4(x) = 5\)[/tex], we need to convert it into an exponential form to find the value of [tex]\(x\)[/tex]. Here’s the step-by-step solution:
1. Understanding the logarithmic equation:
- The equation [tex]\(\log_4(x) = 5\)[/tex] means that the logarithm of [tex]\(x\)[/tex] with base 4 is equal to 5.
2. Converting to exponential form:
- The general form of a logarithmic equation [tex]\(\log_b(a) = c\)[/tex] can be rewritten in exponential form as [tex]\(b^c = a\)[/tex].
- In this case, [tex]\(b = 4\)[/tex], [tex]\(a = x\)[/tex], and [tex]\(c = 5\)[/tex].
3. Rewrite the equation:
- Applying the exponential form conversion: [tex]\(4^5 = x\)[/tex].
4. Calculate the value:
- We now need to determine the value of [tex]\(4^5\)[/tex].
- [tex]\(4^5\)[/tex] means 4 multiplied by itself 5 times: [tex]\(4 \times 4 \times 4 \times 4 \times 4\)[/tex].
5. Result:
- [tex]\(4^5 = 1024\)[/tex].
Therefore, the value of [tex]\(x\)[/tex] is:
[tex]\[ x = 1024 \][/tex]
