Simplifying Logarithmic Expressions: 1/2 log 5 × 5 log 4 × 2 log 1/8
In this article, we will explore how to simplify logarithmic expressions, specifically the given expression:
$\frac{1}{2} \log 5 × 5 \log 4 × 2 \log \frac{1}{8}$
Properties of Logarithms
Before we dive into simplifying the expression, let's recall some important properties of logarithms:
- Product Rule: $\log a × \log b = \log (ab)$
- Power Rule: $\log a^b = b \log a$
- Change of Base Formula: $\log_a b = \frac{\log_k b}{\log_k a}$
Simplifying the Expression
Now, let's break down the given expression into smaller parts and simplify each term:
Term 1: 1/2 log 5
Using the Power Rule, we can rewrite the term as:
$\frac{1}{2} \log 5 = \log 5^{\frac{1}{2}} = \log \sqrt{5}$
Term 2: 5 log 4
We can simplify this term by rewriting it as:
$5 \log 4 = \log 4^5 = \log 1024$
Term 3: 2 log 1/8
Using the Power Rule again, we can rewrite this term as:
$2 \log \frac{1}{8} = \log \left(\frac{1}{8}\right)^2 = \log \frac{1}{64} = -\log 64$
Combining the Terms
Now, let's combine the simplified terms:
$\log \sqrt{5} × \log 1024 × -\log 64$
Using the Product Rule, we can rewrite the expression as:
$\log (\sqrt{5} × 1024 × \frac{1}{64})$
Simplifying further, we get:
$\log \frac{\sqrt{5} × 1024}{64}$
Final Answer
The final simplified expression is:
$\log \frac{5\sqrt{5}}{4}$
In conclusion, we have successfully simplified the given logarithmic expression using the properties of logarithms.
