/*! 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 10 $$ \text { Write an equivalent... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 10

$$ \text { Write an equivalent logarithmic equation. } $$ $$ e^{t}=p $$

Short Answer

Expert verified
The equivalent logarithmic equation is \(t = ln(p)\).

Step by step solution

01

- Identify the components of the exponential equation

In the given exponential equation, the base is the constant \(e\), the exponent (or the power) is \(t\), and the result is \(p\). So the equation is \(e^{t} = p\).
02

- Recall the logarithm definition

The logarithm definition states that for any positive number \(a\), \(a^b = c\) can be rewritten as \(b = \log_{a}(c)\). Here, \(a\) is the base, \(b\) is the exponent, and \(c\) is the result.
03

- Apply the logarithm definition to the equation

Using the logarithm definition, the given equation \(e^{t} = p\) can be rewritten in logarithmic form as \(t = \log_{e}(p)\).
04

- Simplify the logarithmic expression

The natural logarithm \(\log_{e}(p)\) is commonly denoted as \(ln(p)\). Hence, \(t = ln(p)\).

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 Equations
Exponential equations are mathematical expressions in which a variable appears in the exponent. These equations take the form:
  • baseexponent = result
Exponential equations are significant because they model many real-world phenomena, such as population growth and radioactive decay.

In the exercise, the given exponential equation is e^{t} = p.
Here:
  • The base is the constant number 'e' (approximately 2.718).
  • The exponent is the variable 't'.
  • The result is 'p'.
To transform this equation into a logarithmic form, you'll need a solid understanding of the logarithm definition.
Logarithm Definition
The logarithm, abbreviated as 'log,' is the inverse operation to exponentiation.
In simple terms, if you know the outcome of an exponential equation and its base, you can use logarithms to find the exponent.
Consider the general form of the exponential equation: be = r, where 'b' is the base, 'e' is the exponent, and 'r' is the result.

This can be rewritten in logarithmic form as: logb(r) = e

Here,
  • 'log' is the logarithm operation.
  • 'b' is the base of the logarithm (same as the base of the exponential).
  • 'r' is the result of the exponential equation, known as the argument of the logarithm.
  • 'e' is the exponent in the exponential form, now isolated on one side of the equation in logarithmic form.
In our exercise, the equation e^{t} = p uses the base 'e.' It can be written in logarithmic form as loge(p) = t.
Natural Logarithm
The natural logarithm is a special type of logarithm with the base 'e', where 'e' is Euler's number (approximately 2.718).
The natural logarithm is commonly denoted as 'ln'.
It is written as: ln(x).

In our exercise, once we rewrite the exponential equation e^{t} = p into its equivalent logarithmic form: loge(p) = t , you can further simplify it by recognizing that loge(p) is just ln(p).

Therefore, the final equivalent logarithmic equation becomes: t = ln(p).

Natural logarithms are frequently used in higher mathematics, including calculus and complex analysis, for their convenient properties and because they pop up naturally in many growth and decay processes.

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

Double Declining-Balance Depreciation. An office machine is purchased for \(\$ 5200 .\) Under certain assumptions, its salvage value \(V\) depreciates according to a method called double declining balance, basically \(80 \%\) each year, and is given by $$ V(t)=\$ 5200(0.80)^{\ell} $$ where \(t\) is the time, in years. a) Find \(V^{\prime}(t)\). b) Interpret the meaning of \(\mathbf{V}^{\prime}(t)\).

Graph the function. Then analyze the graph using calculus. $$ f(x)=e^{-(1 / 2) x} $$

Differentiate. $$ y=x^{3} \log _{8} x $$

In an experiment where the water temperature is \(25^{\circ} \mathrm{C}\), the probability that a tadpole is captured by notonectids (aquatic insects) is $$ P=1-\frac{1}{1+e^{-5.297+31.669 m}} $$ where \(m\) is the tadpole's mass in grams. Suppose that when the tadpole has a mass of \(0.2 \mathrm{~g}\), it is growing at a rate of \(0.1 \mathrm{~g} / \mathrm{wk}\). How fast is the probability of capture increasing?7

Differentiate. $$ y=\frac{e^{x}}{x^{2}+1} $$
See all solutions

Recommended explanations on Biology Textbooks

Biological Structures

Read Explanation

Ecosystems

Read Explanation

Cell Cycle

Read Explanation

Cells

Read Explanation

Energy Transfers

Read Explanation

Cell Communication

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.