To solve the problem of converting the logarithmic expression [tex]\(\log _a b = x\)[/tex] to its equivalent exponential form, we'll go through the following steps:
1. Understand the Logarithmic Definition:
The logarithm definition [tex]\(\log _a b = x\)[/tex] implies that [tex]\(x\)[/tex] is the power to which the base [tex]\(a\)[/tex] must be raised to yield [tex]\(b\)[/tex].
2. Rewrite the Logarithmic Expression in Exponential Form:
According to the property of logarithms, [tex]\(\log _a b = x\)[/tex] is mathematically equivalent to the exponential equation [tex]\(a^x = b\)[/tex]. This means that raising [tex]\(a\)[/tex] to the power of [tex]\(x\)[/tex] gives us [tex]\(b\)[/tex].
3. Compare the Exponential Form to the Given Options:
We need to match [tex]\(a^x = b\)[/tex] with one of the options provided:
- (A) [tex]\(a^x = b\)[/tex]
- (B) [tex]\(a = b^x\)[/tex]
- (C) [tex]\(x^a = b\)[/tex]
- (D) [tex]\(b = a^x\)[/tex]
From this comparison, it is clear that option (A) [tex]\(a^x = b\)[/tex] correctly represents the exponential form of [tex]\(\log _a b = x\)[/tex].
Therefore, the correct answer is:
(A) [tex]\(a^x = b\)[/tex].
