Question

Write log Subscript a Baseline a equals 2 x in exponential form and find x to evaluate log Subscript a Baseline a for any agreater than​0, anot equals1.
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Part 1
The exponential form of the equation log Subscript a Baseline a equals 2 x is
  
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Answer 1

To convert the logarithmic equation \( \log_a a = 2x \) into exponential form, we need to recall the relationship between logarithms and exponents. The expression \( \log_a b = c \) can be rewritten in exponential form as \( a^c = b \).

In this specific case:
- The base \( a \) is raised to the power of \( 2x \),
- The result is \( a \).

So, the exponential form of the equation \( \log_a a = 2x \) is:

\[
a^{2x} = a
\]

Next, we need to find \( x \) to evaluate \( \log_a a \). We can set the exponents equal to each other since the bases are the same (and \( a > 0 \), \( a \neq 1 \)):

\[
2x = 1
\]

To solve for \( x \), divide both sides by 2:

\[
x = \frac{1}{2}
\]

So, the value of \( x \) is \( \frac{1}{2} \). The evaluation of \( \log_a a \) is simply 1 since the logarithm of a number to its own base is always 1:

\[
\log_a a = 1
\]

Thus, the answers are:

1. Exponential form: \( a^{2x} = a \)
2. Value of \( x \): \( \frac{1}{2} \)