Math Problem Statement

Nilai dari ²log 40 + ²log 24 - ²log 600 + ²log 10 adalah . . .

Solution

The problem in the image asks to find the value of the following expression:

log240+log224log2600+log210\log_2 40 + \log_2 24 - \log_2 600 + \log_2 10

We can simplify this using the properties of logarithms:

  1. Logarithm Addition: logb(x)+logb(y)=logb(xy)\log_b(x) + \log_b(y) = \log_b(x \cdot y)
  2. Logarithm Subtraction: logb(x)logb(y)=logb(xy)\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)

Step-by-step solution:

  1. Combine the logarithms that are added: log2(4024)=log2(960)\log_2(40 \cdot 24) = \log_2(960)

  2. Combine the subtraction: log2(960)log2(600)=log2(960600)=log2(85)\log_2(960) - \log_2(600) = \log_2\left(\frac{960}{600}\right) = \log_2\left(\frac{8}{5}\right)

  3. Lastly, add log210\log_2 10: log2(85)+log2(10)=log2(8510)=log2(16)\log_2\left(\frac{8}{5}\right) + \log_2(10) = \log_2\left(\frac{8}{5} \cdot 10\right) = \log_2(16)

Since log216=4\log_2 16 = 4, the final value is:

44

The value of the expression is 4.

Would you like further details or clarifications?

Here are five related questions for further practice:

  1. What is the value of log232log28+log24\log_2 32 - \log_2 8 + \log_2 4?
  2. How do the properties of logarithms simplify expressions involving multiple logarithms?
  3. What is the value of log381\log_3 81?
  4. How do you convert logarithms with different bases?
  5. How do logarithmic properties relate to exponential functions?

Tip: When working with logarithms, remember the properties of logarithms (product, quotient, and power rules) as they are useful in simplifying complex expressions.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Properties of Logarithms

Formulas

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

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 10-12