/*! 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 21 Evaluate each logarithm. Do not ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 21

Evaluate each logarithm. Do not use a calculator. $$\log _{4} 16$$

Short Answer

Expert verified
The value of \( \log _{4} 16 \) is 2.

Step by step solution

01

Understand the problem

We are given the logarithm \( \log _{4} 16 \). Here, 4 is the base and 16 is the number we want to evaluate. According to the logarithmic formula \( \log_b a = n \), where b is the base, a is the number, and n is the result, we need to find a number n such that \( 4^n = 16 \).
02

Evaluate the power

Now we need to find a power \( n \) such that when 4 is raised to that power, we will get 16. In other words, we have to solve \( 4^n = 16 \). By simple observation, we know that \( 4^2 \) equals 16. So, the result of the given logarithm \( \log _{4} 16 \) is 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.

Logarithmic Formula
Logarithms are the inverse operations of exponents, similar to how subtraction is the inverse of addition. The logarithmic formula is expressed as \( \log_b a = n \). This means that in order to evaluate a logarithm, you're finding the number \( n \) such that the base \( b \) raised to the power of \( n \) equals \( a \). For instance, in the problem \( \log_4 16 \), 4 is the base and 16 is the number you're interested in. You need to find the power to which 4 must be raised to get 16.
  • Base (b): This is the number that you'll raise to a power.
  • Result (a): This is the number you get after raising the base to the power.
  • Exponent (n): This determines how many times the base is multiplied by itself to reach the result.
Understanding this relationship makes it easier to evaluate logarithms without needing a calculator.
Exponents
Exponents are a mathematical way to show how many times a number (known as the base) is multiplied by itself. In the expression \( b^n \), \( b \) is the base and \( n \) is the exponent. Using exponents, you can express large numbers compactly, like \( 4^2 = 16 \), indicating that 4 needs to be multiplied by itself once to get 16. Exponential expressions are crucial when working with logarithms, since logarithms determine which exponent you use. Some key points to remember are:
  • If \( n = 0 \), \( b^n = 1 \) for any base \( b \) except zero.
  • If \( n = 1 \), \( b^n = b \).
  • Exponents tell you the power to which the base is raised to get a specific number.
Evaluating Logarithms
The process of evaluating a logarithm involves things like identifying the base and comparing it to the result you want. In the example \( \log_4 16 \), you understand that the base is 4 and the goal is to find the power \( n \) such that \( 4^n = 16 \). To solve this, you can mentally compute what power of 4 gives you 16. In this case, you realize that \( 4^2 = 16 \). Therefore, \( n = 2 \), and thus \( \log_4 16 = 2 \).Identifying powers can be done quickly with practice:
  • Re-write the number in terms of the base if possible.
  • Compare "powers of" from commonly known values like 2, 3, 4, and so on.
  • Always know that the result of a logarithm is the exponent you are looking for.
With the foundational knowledge of logarithmic structures and exponentiation, evaluating logs becomes quite intuitive.

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

Graph \(g(x)=10^{x}\), and then sketch the graph of \(g\) reflected across the line given by \(y=x\)

Escherichia coli (E. coli) is a bacterium that can reproduce at an exponential rate. The \(E\) coli reproduce by dividing. A small number of E. coli bacteria in the large intestine of a human can trigger a serious infection within a few hours. Consider a particular E. coli infection that starts with \(100 E .\) coli bacteria. Each bacterium splits into two parts every half hour. Assuming none of the bacteria die, the size of the \(E .\) coli population after \(t\) hours is given by \(P(t)=100 \cdot 2^{2 t},\) where \(0 \leq t \leq 16\) a. Find \(P(3)\) and \(P(6)\) b. Use a graphing utility to find the time, to the nearest tenth of an hour, it takes for the \(E .\) coli population to number 1 billion.

A cup of coffee is heated to \(180^{\circ} \mathrm{F}\) and placed in a room that maintains a temperature of \(65^{\circ} \mathrm{F}\). The temperature of the coffee after \(t\) minutes is given by \(T(t)=65+115 e^{-0.042 t}\) a. Find the temperature, to the nearest degree, of the coffee 10 minutes after it is placed in the room. b. Use a graphing utility to determine when, to the nearest tenth of a minute, the temperature of the coffee will reach \(100^{\circ} \mathrm{F}\)

The distance \(s\) (in feet) that the object in Exercise 31 will fall in \(t\) seconds is given by \(s=32 t+32\left(e^{-t}-1\right)\) a. Use a graphing utility to graph this equation for \(t \geq 0\) b. Determine, to the nearest 0.1 second, the time it takes the object to fall 50 feet. c. Calculate the slope of the secant line through \((1, s(1))\) and \((2, s(2))\) d. Write a sentence that explains the meaning of the slope of the secant line you calculated in \(c .\)

The retirement account for a graphic designer contains \(\$ 250,000\) on January 1 \(2002,\) and earns interest at a rate of \(0.5 \%\) per month. On February \(1,2002,\) the designer withdraws \(\$ 2000\) and plans to continue these withdrawals as retirement income each month. The value \(V\) of the account after \(x\) months is $$V=400,000-150,000(1.005)^{x}$$ If the designer wishes to leave \(\$ 100,000\) to a scholarship foundation, what is the maximum number of withdrawals (to the nearest month) the designer can make from this account and still have \(\$ 100,000\) to donate?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.