To solve the logarithmic equation [tex]\(\log 478 = a\)[/tex] and convert it into its equivalent exponential form, follow these steps:
1. Understand the logarithmic equation:
The equation [tex]\(\log 478 = a\)[/tex] states that [tex]\(a\)[/tex] is the logarithm of 478 with base 10.
2. Recall the logarithmic and exponential relationship:
The logarithmic equation [tex]\(\log_b(x) = y\)[/tex] is equivalent to the exponential equation [tex]\(b^y = x\)[/tex]. Here, [tex]\(b\)[/tex] is the base of the logarithm, [tex]\(x\)[/tex] is the number for which you are taking the logarithm, and [tex]\(y\)[/tex] is the result.
3. Identify the base, [tex]\(b\)[/tex], the result, [tex]\(y\)[/tex], and the number, [tex]\(x\)[/tex], in the given logarithmic equation:
In [tex]\(\log 478 = a\)[/tex]:
- The base [tex]\(b\)[/tex] is 10 (since [tex]\(\log\)[/tex] without an explicit base typically means base 10).
- The result [tex]\(y\)[/tex] is [tex]\(a\)[/tex].
- The number [tex]\(x\)[/tex] is 478.
4. Convert the logarithmic equation to its exponential form:
Using the relationship [tex]\(\log_b(x) = y\)[/tex] is equivalent to [tex]\(b^y = x\)[/tex], substitute [tex]\(b\)[/tex], [tex]\(x\)[/tex], and [tex]\(y\)[/tex] from the given equation:
- [tex]\(b = 10\)[/tex]
- [tex]\(x = 478\)[/tex]
- [tex]\(y = a\)[/tex]
This gives us:
[tex]\[
10^a = 478
\][/tex]
Hence, the exponential equation equivalent to the logarithmic equation [tex]\(\log 478 = a\)[/tex] is [tex]\(10^a = 478\)[/tex].
