/*! 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 46 Convert to an exponential equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 46

Convert to an exponential equation. \(t=\log _{4} 7\)

Short Answer

Expert verified
The exponential form is \[4^{t} = 7\]

Step by step solution

01

Understand the Definition of Logarithms

The logarithm \(\t = \log_{4} 7\) is asking the question: 'To what power must 4 be raised, to get 7?'
02

Convert to Exponential Form

Using the definition of logarithms, we can rewrite the equation in exponential form as \[4^{t} = 7\]

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.

Logarithms
Logarithms might sound complicated at first, but they are just another way to look at numbers. Let's take the example \(\t = \log_{4} 7\). This means we are asking, 'To what power must 4 be raised to get 7?' The base of the logarithm is 4, and the result we want is 7. Logarithms can help us solve equations where the unknown is an exponent. We write it in the form \(\t = \log_{base} (result)\). In our case, this becomes \(\t = \log_{4} 7\). Each logarithm problem can be converted to a corresponding exponential problem, which we'll explore in the next section.
Exponential Form
To convert logarithmic equations to exponential form, remember this: If \(\log_{b}(a) = c\), then \( b^c = a\). In simpler terms, if \(log_b(a) = c\), you can rewrite it as the base \(b\) raised to power \(c\) giving \(a\). Let's apply this to our example, \(\log_{4} 7 = t\). According to our rule, we can convert it to \(4^t = 7\). In exponential form, it's easier to visually understand that 4 raised to some power \(t\) equals 7. This method makes solving for \(t\) straightforward.
Mathematical Equations
Mathematical equations like logarithms and exponentials are powerful tools. They appear in many areas, from algebra to calculus, and even in real-world applications like computing and science. When converting from logarithmic to exponential form, you essentially change the perspective but not the meaning of the equation. This skill is crucial for solving more complex problems, simplifying equations, and understanding growth patterns. For example, in finance, exponential equations help calculate compound interest. Understanding these fundamentals gives you a solid foundation in mathematics and boosts your problem-solving skills.

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

Increasing CPU Power The central processing unit (CPU) power in computers has increased significantly over the years. The CPU power in Macintosh computers has grown exponentially from 8 MHz in 1984 to \(3400 \mathrm{MHz}\) in 2013 (Source: Apple). The exponential function $$ M(t)=7.91477(1.26698)^{t} $$ where \(t\) is the number of years after \(1984,\) can be used to estimate the CPU power in a Macintosh computer in a given year. Find the CPU power of a Macintosh Performa \(5320 \mathrm{CD}\) in 1995 and of an iMac G6 in \(2009 .\) Round to the nearest one MHz.

Salvage Value. \(\quad\) A restaurant purchased a 72 -in. range with six burners for \(\$ 6982 .\) The value of the range each year is \(85 \%\) of the value of the preceding year. After \(t\) years, its value, in dollars, is given by the exponential function \(V(t)=6982(0.85)^{t}\) a) Graph the function. b) Find the value of the range after \(0,1,2,5,\) and 8 years. c) The restaurant decides to replace the range when its value has declined to \(\$ 1000 .\) After how long will the range be replaced?

In Exercises \(77-80\) : a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or a minimum value and find that value.[ 3.3] $$f(x)=-x^{2}+6 x-8$$

Approximate the point \((s)\) of intersection of the pair of equations. $$y=2.3 \ln (x+10.7), y=10 e^{-0.007 x^{2}}$$

Solve using any method. $$x^{\log x}=\frac{x^{3}}{100}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.