Simplify the expression log_a(√a) + log_a(a^2).
Understand the Problem
The question asks us to simplify the expression involving logarithms. Specifically, it involves the logarithm of a square root and the logarithm of a power, and we will use properties of logarithms to combine them.
Answer
The simplified expression is $ \frac{5}{2} $.
Answer for screen readers
The simplified expression is $ \frac{5}{2} $.
Steps to Solve
- 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:
$$ \log_a(\sqrt{a}) = \log_a(a^{1/2}) $$
- 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:
$$ \log_a(\sqrt{a}) = \frac{1}{2} $$
- 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:
$$ \log_a(a^2) = 2 $$
- Combine the results
Now we combine our results:
$$ \log_a(\sqrt{a}) + \log_a(a^2) = \frac{1}{2} + 2 $$
- Final simplification
Adding these values:
$$ \frac{1}{2} + 2 = \frac{1}{2} + \frac{4}{2} = \frac{5}{2} $$
The simplified expression is $ \frac{5}{2} $.
More Information
This problem illustrates the use of logarithmic properties, including the power rule and how to manipulate roots as exponents. Understanding these properties is essential for simplifying logarithmic expressions.
Tips
- A common mistake is forgetting to apply the power rule correctly when dealing with exponents.
- Misinterpreting the square root in logarithmic terms can also lead to incorrect simplifications. Always rewrite square roots as exponent terms.
