Math Problem Statement

Solve: log2(16)+log(1/1000)

Solution

To solve the expression log2(16)+log(11000)\log_2(16) + \log\left(\frac{1}{1000}\right), let's break it down into two parts:

1. Calculate log2(16)\log_2(16):

  • The base-2 logarithm of 16 can be expressed as: log2(16)=log2(24)\log_2(16) = \log_2(2^4)
  • Since logb(ac)=clogb(a)\log_b(a^c) = c \cdot \log_b(a), we get: log2(16)=4log2(2)=41=4\log_2(16) = 4 \cdot \log_2(2) = 4 \cdot 1 = 4

2. Calculate log(11000)\log\left(\frac{1}{1000}\right):

  • The logarithm of a fraction can be calculated as: log(11000)=log(103)\log\left(\frac{1}{1000}\right) = \log\left(10^{-3}\right)
  • Using the property log(ab)=blog(a)\log(a^b) = b \cdot \log(a), we have: log(103)=3log(10)\log\left(10^{-3}\right) = -3 \cdot \log(10)
  • Since log(10)\log(10) in base 10 is 1, we get: log(11000)=31=3\log\left(\frac{1}{1000}\right) = -3 \cdot 1 = -3

3. Add the two results together:

log2(16)+log(11000)=4+(3)=1\log_2(16) + \log\left(\frac{1}{1000}\right) = 4 + (-3) = 1

So, the solution is 1\boxed{1}.

Would you like to dive deeper into logarithmic properties, or do you have any other questions?

Here are 5 related questions to explore further:

  1. How would you solve log2(32)\log_2(32)?
  2. What is the value of log(1100)\log\left(\frac{1}{100}\right)?
  3. How do logarithm properties apply to solve log5(25)+log(100)\log_5(25) + \log(100)?
  4. Can you evaluate log2(1)\log_2(1) and explain why it results in that value?
  5. What is the inverse function of a logarithm, and how is it used?

Tip: Always remember that logb(1)=0\log_b(1) = 0 for any base bb because any number raised to the power of zero is 1.

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Math Problem Analysis

Mathematical Concepts

Logarithms

Formulas

Properties of logarithms

Theorems

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Suitable Grade Level

High School