/*! 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 14 Change each equation to its loga... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 14

Change each equation to its logarithmic form. Assume \(y>0\) and \(b>0\). $$10^{0}=1$$

Short Answer

Expert verified
The logarithmic form of the equation \(10^0 = 1\) will be \(\log{1} = 0\).

Step by step solution

01

Identify the components in the given equation

The base (b) in the given equation is 10, the exponent (y) is 0, and the result (x) is 1: \(10^0 = 1\).
02

Apply the exponential-logarithmic transformation

Use the transformation rule to convert from exponential form to logarithmic form. In this case, the base (b) is 10, x is 1, and y is 0. Therefore, the logarithmic equation will be \(\log_{10}{1} = 0\).
03

Simplify the Logarithmic Form

In the field of mathematics, \(\log_{10}{1}\) is typically written as \(\log{1}\). Therefore, the simplified form of the equation will be \(\log{1} = 0\).

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.

Exponential to Logarithmic Transformation
Transforming an exponential equation to a logarithmic form is like expressing the same idea in a different language. It's crucial to grasp the components of the equation. Let's think of an exponential equation as having three main parts: the base, the exponent, and the result. For example, in the equation \(10^0 = 1\), the base is 10, the exponent is 0, and the result is 1.

When we convert this expression into its logarithmic form, we rearrange these parts. The logarithmic form places the result as the argument of the log function, the base remains as the base of the logarithm, and the equation equals the exponent. So, for \(10^0 = 1\), this turns into \(\log_{10}{1} = 0\).

This transformation uses the rule that says: if \(b^y = x\), then \(\log_b{x} = y\). Learning this shift makes solving many math problems more manageable.
Logarithmic Equation
A logarithmic equation is simply an equation that involves a logarithm. Once we change an exponential form to a logarithmic form, we often find ourselves dealing with a logarithmic equation.

In our given exercise, after converting \(10^0 = 1\) to \(\log_{10}{1} = 0\), we get a logarithmic equation. Such equations often require us to identify components like the base and the argument.

Understanding these components is key to solving logarithmic equations. For instance, in \(\log_{10}{1} = 0\), the number 1 is the argument of the log, which tells us what number has been logged, and the base is 10, showing which base we use. The equal part, 0 here, tells us what power the base must be raised to, to achieve the argument.
Base of Logarithm
The base of a logarithm is a fundamental element of both exponential and logarithmic equations. It represents the number raised to a power to obtain the argument.

In our example \(\log_{10}{1} = 0\), the base is 10. This means that what we are determining is to what power we need to raise 10 (our base) to get 1, which in this exercise is 0. The choice of base affects the resulting value greatly.

Here are a few important characteristics of base in logarithms:
  • The base must be a positive number greater than zero.
  • The most common bases are 10 (common log), 2 (binary log), and \(e\) (natural log).
  • The base explains the entire stretching or compressing effect on the logarithm's graph.
Understanding the base helps simplify and solve logarithmic equations accurately.

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

Assuming that air resistance is proportional to velocity, the velocity \(v,\) in feet per second, of a falling object after \(t\) seconds is given by \(v=32\left(1-e^{-t}\right)\) a. Graph this equation for \(t \geq 0\) b. Determine algebraically, to the nearest 0.01 second, when the velocity is 20 feet per second. c. Determine the horizontal asymptote of the graph of \(v\) d. the horizontal asymptote in the context of this application.

Involve the factorial function \(x !\), which is defined for whole numbers \(x\) as $$ x !=\left\\{\begin{array}{ll} 1, & \text { if } x=0 \\ x \cdot(x-1) \cdot(x-2) \cdot \cdots \cdot \cdot 3 \cdot 2 \cdot 1, & \text { if } x \geq 1 \end{array}\right. $$ For example, \(3 !=3 \cdot 2 \cdot 1=6\) and \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120\) During the 30 -minute period before a Broadway play begins, the members of the audience arrive at the theater at the average rate of 12 people per minute. The probability that \(x\) people will arrive during a particular minute is given by \(P(x)=\frac{12^{x} e^{-12}}{x !} .\) Find the probability, to the nearest \(0.1 \%\) that a. 9 people will arrive during a given minute. b. 18 people will arrive during a given minute.

Data from Forrester Research suggest that the number of broadband [cable and digital subscriber line (DSL) \(]\) connections to the Internet can be modeled by \(f(x)=1.353(1.9025)^{x},\) where \(x\) is the number of years after January \(1,1998,\) and \(f(x)\) is the number of connections in millions. a. How many broadband Internet connections, to the nearest million, does this model predict will exist on January \(1,2005 ?\) b. According to the model, in what year will the number of broadband connections first reach 300 million? [Hint: Use the intersect feature of a graphing utility to determine the \(x\) -coordinate of the point of intersection of the graphs of \(f(x)\) and \(y=300.1\)

Use a graphing utility to graph each function. If the function has a horizontal asymptote, state the equation of the horizontal asymptote. $$f(x)=-e^{(x-4)}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

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.