/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Evaluate the expression. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

Evaluate the expression. $$ \log \frac{1}{\sqrt{1000}} $$

Short Answer

Expert verified
The value of the expression is \(-\frac{3}{2}\).

Step by step solution

01

Rewrite the Expression

To simplify the expression, let's rewrite the given logarithmic expression: \( \log \frac{1}{\sqrt{1000}} \). This can be rewritten as \( \log (1000^{-1/2}) \) because \( \sqrt{1000} = 1000^{1/2} \) and the reciprocal is the negative exponent: \( \frac{1}{x} = x^{-1} \).
02

Apply the Power Rule of Logarithms

Using the power rule of logarithms, which states that \( \log(a^b) = b \log(a) \), we can rewrite the expression as \(-\frac{1}{2} \cdot \log(1000) \).
03

Evaluate Logarithm of 1000

Recognize that 1000 is \( 10^3 \). By using the property of logarithms, \( \log(a^b) = b \log(a) \), we compute \( \log(1000) = \log(10^3) = 3 \log(10) \). Since \( \log(10) = 1 \) (for common logarithm with base 10), we find that \( \log(1000) = 3 \cdot 1 = 3 \).
04

Calculate the Result

Substitute back the value of \( \log(1000) = 3 \) into our expression from Step 2: \(-\frac{1}{2} \cdot 3 = -\frac{3}{2} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Power Rule of Logarithms
The power rule of logarithms is a fundamental tool that simplifies logarithmic expressions involving exponents. The rule states that if you have a logarithm of a number raised to an exponent, such as \( \log(a^b) \), you can bring the exponent down as a coefficient in front of the logarithm. Mathematically, it's expressed as:
  • \( \log(a^b) = b \cdot \log(a) \)
This transforms the problem into a simpler multiplication problem rather than dealing with an exponent directly.
For example, in our exercise, we simplified \( \log(1000^{-1/2}) \) to \( -\frac{1}{2} \cdot \log(1000) \). This conversion helps because we can focus on evaluating the simpler logarithm, \( \log(1000) \), separately.
Properties of Logarithms
Logarithms have several properties that make them a powerful tool in mathematics, enabling the simplification and solving of complex equations.
Here are some key properties:
  • Product Rule: \( \log(ab) = \log(a) + \log(b) \)
  • Quotient Rule: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)
  • Power Rule: \( \log(a^b) = b \cdot \log(a) \)
  • Change of Base Formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)
In our example, the power rule was key in simplifying the expression \( \log(1000^{-1/2}) \). We used the understanding that \( 1000^{-1/2} \) implies squaring and inverting the base, which we effectively handled thanks to these properties.
Understanding these properties allows handling different forms of expressions simply by applying the right rule.
Common Logarithm
A common logarithm is a logarithm with the base of 10 and is often denoted simply as \( \log(x) \) rather than \( \log_{10}(x) \). It's commonly used in many scientific and engineering disciplines due to the simplicity when dealing with powers of 10.
For example, the common logarithm of 10 is 1 because 10 raised to the power of 1 equals 10: \( \log(10) = 1 \). This property made it straightforward in our exercise to compute \( \log(1000) \) because 1000 is \( 10^3 \), leading to:
  • \( \log(10^3) = 3 \cdot \log(10) = 3 \cdot 1 = 3 \)
Recognizing that a common logarithm simplifies evaluation encourages using base 10 for routine calculations, especially when numbers naturally scale as powers of 10.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the solution of the exponential equation, correct to four decimal places. $$ 2^{3 x+1}=3^{x-2} $$

Solve the logarithmic equation for \(x\) $$ 2-\ln (3-x)=0 $$

Solve the equation. $$ e^{x}-12 e^{-x}-1=0 $$

A hot bowl of soup is served at a dinner party. It starts to cool according to Newton鈥檚 Law of Cooling so that its temperature at time \(t\) is given by $$T(t)=65+145 e^{-0.05 t}$$ where \(t\) is measured in minutes and \(T\) is measured in \(^{\circ} \mathrm{F}\) (a) What is the initial temperature of the soup? (b) What is the temperature after 10 \(\mathrm{min}\) ? (c) After how long will the temperature be \(100^{\circ} \mathrm{F} ?\)

Radium-221 has a half-life of 30 s. How long will it take for 95% of a sample to decay?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.