Math Problem Statement

Nilai dari 2log 20 + 2log45 - 2log180 - 2log40 adalah . . .

Solution

The problem asks for the value of:

2log⁡20+2log⁡45−2log⁡180−2log⁡40{}^2\log 20 + {}^2\log 45 - {}^2\log 180 - {}^2\log 40

We can simplify this expression using the properties of logarithms. Recall that:

  1. log⁡b(x⋅y)=log⁡bx+log⁡by\log_b (x \cdot y) = \log_b x + \log_b y (product rule).
  2. log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y (quotient rule).

Step 1: Combine the logs

Using the quotient rule for logarithms:

= {}^2\log \left( \frac{20 \cdot 45}{180 \cdot 40} \right)$$ ### Step 2: Simplify the fraction First, compute the product in the numerator and denominator: $$\frac{20 \cdot 45}{180 \cdot 40} = \frac{900}{7200} = \frac{1}{8}$$ ### Step 3: Apply the logarithm Now we apply the logarithm to the simplified fraction: $${}^2\log \left(\frac{1}{8}\right)$$ We know that $$\frac{1}{8} = 2^{-3}$$, so: $${}^2\log (2^{-3}) = -3$$ Thus, the value of the expression is $$-3$$. The correct answer is **B. -3**. --- Would you like more details or have any questions? Here are some related questions to further your understanding: 1. How do logarithmic properties help simplify complex expressions? 2. What is the general rule for simplifying logarithms of fractions? 3. How would the answer change if one of the logarithms had a different base? 4. Can you apply the same properties for natural logarithms (base $$e$$)? 5. How can you express $$\log_b (x^n)$$ using logarithmic properties? **Tip**: Always look for opportunities to apply logarithmic properties (such as product, quotient, and power rules) to simplify expressions before calculating.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Logarithmic properties

Formulas

log_b(x * y) = log_b x + log_b y
log_b(x / y) = log_b x - log_b y
log_b(x^n) = n * log_b(x)

Theorems

Quotient Rule for Logarithms
Product Rule for Logarithms

Suitable Grade Level

Grades 10-12

Related Recommendation

Logarithmic Expression with Base 2: Simplifying 2log20 + 2log45 - 2log180 - 2log40

Simplifying Logarithmic Expression: log 225 + 8 log 6 - 3 log 121

Simplifying Logarithmic Expressions: log 48 + 5log 50 - 2log 3 - 5log 2

Calculate 2log 48 - 2log 3 Using Logarithmic Properties

Simplifying Logarithmic Expression with Bases 5, 2, and 3