Simplify the expression $\log_a{\sqrt{a}} + \log_a{(a^2)}$
Understand the Problem
The question asks us to simplify the logarithmic expression (\log_a{\sqrt{a}} + \log_a{(a^2)}). We will use properties of logarithms to combine and simplify these terms.
Answer
$\frac{5}{2}$
Answer for screen readers
$\frac{5}{2}$
Steps to Solve
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Rewrite the square root as an exponent Rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$. $$\log_a{\sqrt{a}} + \log_a{(a^2)} = \log_a{a^{\frac{1}{2}}} + \log_a{(a^2)}$$
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Apply the power rule of logarithms Use the property $\log_b{x^y} = y\log_b{x}$ to simplify the terms. $$\log_a{a^{\frac{1}{2}}} + \log_a{(a^2)} = \frac{1}{2}\log_a{a} + 2\log_a{a}$$
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Simplify $\log_a{a}$ Since $\log_a{a} = 1$, substitute 1 for $\log_a{a}$. $$\frac{1}{2}\log_a{a} + 2\log_a{a} = \frac{1}{2}(1) + 2(1) = \frac{1}{2} + 2$$
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Add the constants Add the two constants together. $$\frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2}$$
$\frac{5}{2}$
More Information
The logarithm $\log_a{x}$ answers the question, "To what power must $a$ be raised to produce $x$?" The expression simplifies to a constant, $\frac{5}{2}$. This means that $a^{\frac{5}{2}} = \sqrt{a} \cdot a^2$.
Tips
A common mistake is not remembering the power rule of logarithms, or not realizing that $\log_a{a} = 1$. Another mistake may involve arithmetical errors when adding the fractions.
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