Simplify the expression log_a(√a) + log_a(a^2).

Steps to Solve

1. Rewrite the square root as an exponent

The square root of $a$ can be expressed as an exponent:

$$ \sqrt{a} = a^{1/2} $$

Thus, we have:

$$ \log_a(\sqrt{a}) = \log_a(a^{1/2}) $$

2. Apply the power rule of logarithms

Using the power rule of logarithms, which states that $\log_b(m^n) = n \cdot \log_b(m)$, we can simplify:

$$ \log_a(a^{1/2}) = \frac{1}{2} \cdot \log_a(a) $$

Since $\log_a(a) = 1$, this simplifies to:

$$ \log_a(\sqrt{a}) = \frac{1}{2} $$

3. Rewrite the logarithm of the power

Next, simplify the second term:

$$ \log_a(a^2) $$

Using the power rule again:

$$ \log_a(a^2) = 2 \cdot \log_a(a) $$

Which simplifies to:

$$ \log_a(a^2) = 2 $$

4. Combine the results

Now we combine our results:

$$ \log_a(\sqrt{a}) + \log_a(a^2) = \frac{1}{2} + 2 $$

5. Final simplification

Adding these values:

$$ \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} $$