Critical thinking questions: $$ \begin{array}{ll} ext { 25) } 2(\log 2 x-\log y)-(\log 3+2 \log 5) & ext { 26) } \log x \cdot \log 2\end{array} $$

Question

Critical thinking questions: $$ \begin{array}{ll}	ext { 25) } 2(\log 2 x-\log y)-(\log 3+2 \log 5) & 	ext { 26) } \log x \cdot \log 2\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\log_{10}{\left(x\right)}\times \log_{10}{\left(2\right)}$$
- step1: Multiply the terms:
  $$\log_{10}{\left(2\right)}\times \log_{10}{\left(x\right)}$$
Calculate or simplify the expression $$ 2*(\log(2)*\log(x)-\log(y))-(\log(3)+2*\log(5)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$2\left(\log_{10}{\left(2\right)}\times \log_{10}{\left(x\right)}-\log_{10}{\left(y\right)}\right)-\left(\log_{10}{\left(3\right)}+2\log_{10}{\left(5\right)}\right)$$
- step1: Add the terms:
  $$2\log_{10}{\left(\frac{x^{\log_{10}{\left(2\right)}}}{y}\right)}-\left(\log_{10}{\left(3\right)}+2\log_{10}{\left(5\right)}\right)$$
- step2: Add the terms:
  $$2\log_{10}{\left(\frac{x^{\log_{10}{\left(2\right)}}}{y}\right)}-\log_{10}{\left(75\right)}$$
Let's break down the solutions for the two critical thinking questions step by step.

### Question 25:
We need to simplify the expression:
$$
2(\log 2 x - \log y) - (\log 3 + 2 \log 5)
$$

1. **Apply the properties of logarithms**:
   - The property $$\log a - \log b = \log \left(\frac{a}{b}\right)$$ allows us to combine the logarithms.
   - The property $$n \log a = \log(a^n)$$ allows us to handle the coefficients.

2. **Rewrite the expression**:
   $$
   2(\log(2) + \log(x) - \log(y)) - (\log(3) + 2\log(5))
   $$
   becomes:
   $$
   2(\log\left(\frac{2x}{y}\right)) - \log(3) - \log(25)
   $$

3. **Combine the logarithms**:
   $$
   2\log\left(\frac{2x}{y}\right) - \log(75)
   $$
   can be rewritten as:
   $$
   \log\left(\left(\frac{2x}{y}\right)^2\right) - \log(75)
   $$
   which simplifies to:
   $$
   \log\left(\frac{(2x)^2}{75y^2}\right)
   $$

Thus, the final simplified form is:
$$
2\log_{10}\left(\frac{x^{\log_{10}(2)}}{y}\right) - \log_{10}(75)
$$

### Question 26:
We need to simplify the expression:
$$
\log x \cdot \log 2
$$

This expression is already in a simplified form. It represents the product of the logarithm of $$x$$ and the logarithm of $$2$$.

Thus, the final simplified form is:
$$
\log_{10}(2) \times \log_{10}(x)
$$

### Summary of Results:
- **Question 25**: $$2\log_{10}\left(\frac{x^{\log_{10}(2)}}{y}\right) - \log_{10}(75)$$
- **Question 26**: $$\log_{10}(2) \times \log_{10}(x)$$
**Question 25** simplifies to $$2\log_{10}\left(\frac{x^{\log_{10}(2)}}{y}\right) - \log_{10}(75)$$. **Question 26** simplifies to $$\log_{10}(2) \times \log_{10}(x)$$.

Critical thinking questions:

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Solución

Question 25:

Question 26:

Summary of Results:

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