/*! 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 75 Which real number \(x\) satisfie... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 75

Which real number \(x\) satisfies (a) \(\log _{4} x=-2\) ? (b) \(\log _{1 / 3} x=-3 ?\) (c) \(\log _{10} x=-2\) ?

Short Answer

Expert verified
(a) \(x = \frac{1}{16}\); (b) \(x = 27\); (c) \(x = \frac{1}{100}\).

Step by step solution

01

Understand Logarithm Properties

Recall the fundamental property of logarithms: If \( \log_b a = c \), then \( b^c = a \). This property will be used to find the values of \( x \) in each of the problems given.
02

Solve Part (a)

We have \( \log _{4} x=-2 \). Applying the property \( b^c = a \), we have \( 4^{-2} = x \). Thus, \( x = 4^{-2} \), which simplifies to \( x = \frac{1}{16} \).
03

Solve Part (b)

We have \( \log _{1/3} x = -3 \). Using the property \( b^c = a \), we get \( \left( \frac{1}{3} \right)^{-3} = x \). This simplifies to \( x = 3^3 = 27 \).
04

Solve Part (c)

We have \( \log _{10} x = -2 \). According to the property \( b^c = a \), it follows that \( 10^{-2} = x \). This simplifies to \( x = \frac{1}{100} \).

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.

Properties of Logarithms
Logarithms have several properties that make them incredibly useful in mathematics, especially when solving equations. One crucial property is the relationship between a logarithm and exponents: if you have a log in the form of \( \log_b a = c \), it tells you that \( b^c = a \). This property provides a powerful tool for converting between logarithmic and exponential forms, enabling us to solve for unknown values. Other useful properties include:
  • Logarithm of a product: \( \log_b (mn) = \log_b m + \log_b n \).
  • Logarithm of a quotient: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \).
  • Logarithm of a power: \( \log_b (m^n) = n \log_b m \).
These properties allow for the simplification and manipulation of complex logarithmic expressions, making them easier to work with. Understanding these properties is key to mastering logarithms.
Logarithmic Equations
Logarithmic equations involve logarithms and are equations where the unknown variable is inside a logarithm. To solve these types of equations, it's often helpful to use the property that relates logarithms to exponents. For instance, in the equation \( \log_b x = y \), you can rewrite it as \( b^y = x \) to solve for \( x \). This transformation is essential because it converts the equation into a form that many students find easier to work with: an exponential equation.When solving logarithmic equations:
  • Always check if you can use the basic property \( b^c = a \) to rewrite the equation.
  • Be vigilant about the domain of the logarithm, ensuring that the arguments are positive since logarithms of non-positive numbers are undefined.
  • If using properties results in multiple solutions, verify each one to ensure they fit within the allowed domain.
Logarithmic equations may initially seem daunting, but breaking them down using fundamental properties simplifies the process and provides insight into their solutions.
Base of a Logarithm
The base of a logarithm is a critical component that determines the form and properties of the logarithm itself. In a logarithmic expression like \( \log_b a \), \( b \) is the base. Common bases include 10 (common logarithms), \( e \) (natural logarithms), and 2 (binary logarithms), but any positive number except 1 can be a base.Understanding the impact of the base is essential:
  • Logarithms with base 10 are used frequently in science and engineering because many products come with powers of 10.
  • Natural logarithms, which use base \( e \), are crucial in calculus and natural growth processes because \( e \) is the rate of continuous growth.
  • Choosing the appropriate base can simplify calculations, especially if you're dealing with exponential growth or decay that naturally aligns with powers of a certain base.
When working with logarithms, always pay attention to the base, as different bases change the results and may require different approaches to solve equations efficiently.

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

Suppose that \(N(t)\) denotes a population size at time \(t\) and satisfies the equation $$ N(t)=2 e^{3 t} \quad \text { for } t \geq 0 $$ (a) If you graph \(N(t)\) as a function of \(t\) on a semilog plot, a straight line results. Explain why. (b) Graph \(N(t)\) as a function of \(t\) on a semilog plot, and determine the slope of the resulting straight line.

Find the following numbers on a number line that is on a logarithmic scale (base 10\()\) : (i) \(10^{-3}, 2 \times 10^{-3}, 3 \times 10^{-3}\) (ii) \(10^{-1}, 2 \times 10^{-1}, 3 \times 10^{-1}\) (iii) \(10^{2}, 2 \times 10^{2}, 3 \times 10^{2}\) (b) From your answers to (a), how many units (on a logarithmic scale) is (i) \(2 \times 10^{-3}\) from \(10^{-3}\) (ii) \(2 \times 10^{-1}\) from \(10^{-1}\) and (iii) \(2 \times 10^{2}\) from \(10^{2}\) ? (c) From your answers to (a), how many units (on a logarithmic scale) is (i) \(3 \times 10^{-3}\) from \(10^{-3}\) (ii) \(3 \times 10^{-1}\) from \(10^{-1}\) and (iii) \(3 \times 10^{2}\) from \(10^{2}\) ?

Find the following numbers on a number line that is on a logarithmic scale (base 10\(): 10^{2}, 10^{-3}, 10^{-4}, 10^{-7}\), and \(10^{-10}\). (b) Can you find 0 on a number line that is on a logarithmic scale? (c) Can you find negative numbers on a number line that is on a logarithmic scale?

Hummingbird Flight Hummingbird wing-beat frequency decreases as bird mass increases. Altshuler et al. (2010) made the following measurements of bird size (measured in mass, \(B\), in \(g\) ) and wind beat frequency (frequency, \(f\), in \(\mathrm{Hz}\) ). \begin{tabular}{lcc} \hline & & Wing Beat \\ Species & Body Mass (B, & Frequency ( \(\boldsymbol{f},\), \\ & Measured in g) & Measured in Hz) \\ \hline Giant hummingbird & \(22.025\) & \(14.99\) \\ Volcano hummingbird & \(2.708\) & \(43.31\) \\ Blue-mantled thornbill & \(6.000\) & \(29.27\) \\ \hline \end{tabular} Assume that there is a power-law dependence of \(f\) upon \(B\) : $$ f=b B^{a} $$ for some constants \(a\) and \(b\). By plotting \(\log f\) against \(\log B\), estimate the parameters \(a\) and \(b\).

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=2 x^{5} $$
See all solutions

Recommended explanations on Biology Textbooks

Cell Communication

Read Explanation

Responding to Change

Read Explanation

Ecological Levels

Read Explanation

Bioenergetics

Read Explanation

Ecology

Read Explanation

Organ Systems

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.