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Chapter 12: Problem 33

Simplify. $$ \log _{1000} 100 $$

Short Answer

Expert verified
\( \text{\log}_{1000} 100 = \frac{2}{3} \)

Step by step solution

01

Understand the Logarithm Definition

The expression \(\text{\log}_{1000} 100\) can be read as 'the logarithm of 100 with base 1000.' This means we want to find the power to which 1000 must be raised to get 100.
02

Change the Base

Use the change of base formula: \(\text{\log}_a b = \frac{\log_c b}{\log_c a}\). Choosing the common logarithm (base 10): \(\text{\log}_{1000} 100 = \frac{\log_{10} 100}{\log_{10} 1000}\).
03

Calculate Logarithms

Compute the individual logarithms using base 10: \(\text{\log}_{10} 100 = 2\) because \({10}^2 = 100\) and \(\text{\log}_{10} 1000 = 3\) because \({10}^3 = 1000\).
04

Divide the Logarithms

Substitute the values back into the formula: \(\text{\log}_{1000} 100 = \frac{2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Change of Base Formula
Logarithms help us solve for exponents in equations. Sometimes, we need to convert logarithms from one base to another.
This is where the change of base formula comes in. It is written as \( \text{log}_{a} b = \frac{\text{log}_c b}{\text{log}_c a} \).
It lets us convert a logarithm to any base we prefer, often making calculations easier.
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Most popular questions from this chapter

Simplify. $$ a^{12} \cdot a^{6} $$

How would you explain to a classmate why \(\log _{2} 5=\log 5 / \log 2\) and \(\log _{2} 5=\ln 5 / \ln 2 ?\)

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