/*! 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 7 Solve each exponential equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

Solve each exponential equation. Express irrational solutions as decimals correct to the nearest thousandth. $$\left(\frac{1}{2}\right)^{x}=5$$

Short Answer

Expert verified
x 鈮� -2.321

Step by step solution

01

Rewrite the equation

Rewrite the given equation \(\bigg(\frac{1}{2}\bigg)^x = 5\) in logarithmic form to make it easier to solve for \(x\).
02

Apply logarithms

Take the natural logarithm (ln) on both sides of the equation:\( \text{ln}\bigg(\bigg(\frac{1}{2}\bigg)^x\bigg) = \text{ln}(5) \)
03

Use logarithm properties

Use the power rule of logarithms, which states \( \text{ln}(a^b) = b \text{ln}(a) \), to move \(x\) in front of the natural logarithm:\( x \text{ln}\bigg(\frac{1}{2}\bigg) = \text{ln}(5) \)
04

Solve for x

Isolate \(x\) by dividing both sides of the equation by \( \text{ln}\bigg(\frac{1}{2}\bigg) \):\( x = \frac{\text{ln}(5)}{\text{ln}\bigg(\frac{1}{2}\bigg)} \)
05

Calculate the logarithms

Use a calculator to find the approximate values of the logarithms:\(\text{ln}(5) \approx 1.609 \) and \(\text{ln}\bigg(\frac{1}{2}\bigg) \approx -0.693 \)
06

Find the value of x

Divide the values to find \(x\):\( x \approx \frac{1.609}{-0.693} \approx -2.321 \)

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 form
To effectively solve exponential equations, converting them into logarithmic form is a helpful strategy. Logarithmic form is another way to write an equation and makes certain kinds of mathematical operations easier. For the equation \(\bigg(\frac{1}{2}\bigg)^x = 5\), we can rewrite it in logarithmic form as \(\text{log}_{\frac{1}{2}}(5) = x\). This says that \(\frac{1}{2}\) raised to the power of \(x\) equals \(5\). Rewriting the equation in this form is essential for further simplifications.
natural logarithm
A natural logarithm, denoted as \( \text{ln} \), is a logarithm with the base \( e\), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. We often use the natural logarithm because it has many convenient mathematical properties. For our equation, \( \text{ln}\bigg(\bigg(\frac{1}{2}\bigg)^x\bigg) = \text{ln}(5) \), applying the natural logarithm simplifies handling the exponential part.
power rule of logarithms
The power rule of logarithms is a key property that helps to solve equations involving exponents. It states that \( \text{ln}(a^b) = b \text{ln}(a) \). It allows us to bring the exponent to the front of the logarithm. In our problem, we get \( x \text{ln}\bigg(\frac{1}{2}\bigg) = \text{ln}(5) \), making it easier to solve for \( x \) by isolating it.
irrational solutions
When solving logarithmic or exponential equations, we sometimes get irrational solutions. These solutions can't be expressed as exact fractions but instead as non-repeating and non-terminating decimals. For our given problem, \( x = \frac{\text{ln}(5)}{\text{ln}\bigg(\frac{1}{2}\bigg)} \), results in an approximate value: \( x \) roughly equals -2.321 when the logarithms are calculated and divided. This is an irrational solution, which we must often approximate to a certain decimal place for practical use.

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

World Population Growth since 2000 , world population in millions closely fits the exponential function $$ y=6084 e^{00120 x} $$ where \(x\) is the number of years since 2000 . (Source: U.S. Census Bureau.) (a) The world population was about 6853 million in 2010 . How closely does the function approximate this value? (b) Use this model to predict the population in 2020 . (c) Use this model to predict the population in 2030 . (d) Explain why this model may not be accurate for 2030 .

For individual or collaborative investigation (Exercises \(117-122\) ) Assume \(f(x)=a^{x}\), where \(a>1 .\) Work these exercises in order. If \(\left.a=e, \text { what is the equation for } y=f^{-1}(x) ? \text { (You need not solve for } y .\right)\)

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. $$\log _{1 / 3} 2$$

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts \((a)-(c)\) Given \(g(x)=e^{x},\) find (a) \(g(\ln 4) \quad\) (b) \(g\left(\ln \left(5^{2}\right)\right)\) (c) \(g\left(\ln \left(\frac{1}{\varepsilon}\right)\right)\)

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=x^{3}+1$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.