Simplify the terms on the left-hand side (LHS) using the logarithm power rule, n a = a^n.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Simplify the terms on the left-hand side (LHS) using the logarithm power rule, $n \log a = \log a^n$.
The LHS is $2\log 4 + \frac{1}{2}\log 25 - \log 20$.
$$2\log 4 = \log 4^2 = \log 16$$
$$\frac{1}{2}\log 25 = \log 25^{1/2} = \log \sqrt{25} = \log 5$$
Substitute these back into the LHS:
$$\text{LHS} = \log 16 + \log 5 - \log 20$$
Step 2: Combine the terms using the logarithm product rule, $\log a + \log b = \log (ab)$.
$$\log 16 + \log 5 = \log (16 \times 5) = \log 80$$
Now the LHS is:
$$\text{LHS} = \log 80 - \log 20$$
Step 3: Combine the remaining terms using the logarithm quotient rule, $\log a - \log b = \log \left(\frac{a}{b}\right)$.
$$\text{LHS} = \log \left(\frac{80}{20}\right) = \log 4$$
Step 4: Simplify the right-hand side (RHS) using the logarithm power rule.
The RHS is $2\log 2$.
$$\text{RHS} = \log 2^2 = \log 4$$
Step 5: Compare the simplified LHS and RHS.
We found $\text{LHS} = \log 4$ and $\text{RHS} = \log 4$.
Since $\text{LHS} = \text{RHS}$, the statement is shown to be true.
$$\boxed{\text{LHS} = \log 4 = \text{RHS}}$$
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