Which expressions are equivalent to the given expression?

Ask by Herrera Ellis. in the United States

Dec 23,2024

Question

Which expressions are equivalent to the given expression? $$ 5 \log _{10} x+\log _{10} 20-\log _{10} 10 $$

UpStudy AI · Accepted Answer

To determine which expressions are equivalent to the given expression:

$$
5 \log_{10} x + \log_{10} 20 - \log_{10} 10
$$

we can apply the properties of logarithms step-by-step.

1. **Apply the Power Rule**:  
   The power rule states that $$ a \log_b c = \log_b c^a $$.  
   $$
   5 \log_{10} x = \log_{10} x^5
   $$
   So, the expression becomes:
   $$
   \log_{10} x^5 + \log_{10} 20 - \log_{10} 10
   $$

2. **Combine Using the Product Rule**:  
   The product rule states that $$ \log_b A + \log_b B = \log_b (A \times B) $$.  
   $$
   \log_{10} x^5 + \log_{10} 20 = \log_{10} (x^5 \times 20)
   $$
   Now the expression is:
   $$
   \log_{10} (20x^5) - \log_{10} 10
   $$

3. **Apply the Quotient Rule**:  
   The quotient rule states that $$ \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right) $$.  
   $$
   \log_{10} (20x^5) - \log_{10} 10 = \log_{10} \left(\frac{20x^5}{10}\right)
   $$
   Simplify the fraction:
   $$
   \frac{20}{10} = 2 \quad \Rightarrow \quad \log_{10} (2x^5)
   $$

**Final Equivalent Expression:**
$$
\log_{10} (2x^5)
$$

**Other Equivalent Forms:**
- Using the product rule in reverse: $$ \log_{10} 2 + 5 \log_{10} x $$
- Using exponential form: $$ \log_{10} (2 \cdot x^5) $$

All these forms are mathematically equivalent to the original expression.

**Answer:**  
After simplifying, an equivalent expression is log₁₀ ( 2 · x⁵ ). Thus,

log₁₀(2x⁵)
An equivalent expression is $$ \log_{10}(2x^5) $$.

Which expressions are equivalent to the given expression?

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