/*! 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 42 For Problems \(41-50\), solve ea... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 42

For Problems \(41-50\), solve each equation. $$ \log _{2} x=5 $$

Short Answer

Expert verified
The solution to \( \log_{2} x = 5 \) is \( x = 32 \).

Step by step solution

01

Understand the equation

The given equation is \( \log_{2} x = 5 \). This means that the logarithm of \( x \) with base 2 equals 5. We need to find the value of \( x \).
02

Convert from logarithmic to exponential form

Recall that the expression \( \log_{b} a = c \) is equivalent to saying \( b^{c} = a \). Apply this principle: \( \log_{2} x = 5 \) becomes \( 2^{5} = x \).
03

Calculate the power

Compute \( 2^{5} \): \[ 2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32 \].
04

Write the solution

Thus, the value of \( x \) is 32, so \( x = 32 \).

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.

Understanding Logarithms
Logarithms are a way to solve equations involving exponents. If you have an equation like \( \log_{b} x = y \), it means you're trying to find what power you need to raise the base \( b \) to in order to get \( x \). In simpler terms, it's asking: "What power of \( b \) equals \( x \)?"
  • The number \( b \) is the base of the logarithm.
  • The number \( x \) is what the logarithm equals when \( b \) is raised to a certain power.
  • The result, \( y \), is the power that \( b \) needs.
This makes logarithms incredibly useful for solving equations where the unknown is an exponent. By converting logarithmic equations to their exponential form, they become much easier to tackle.
Exponential Form Conversion
To solve logarithmic equations, converting them to exponential form can be very helpful. The general rule is if you have \( \log_{b} a = c \), it can be rewritten as \( b^{c} = a \).
  • The base \( b \) is the same in both forms.
  • The result \( a \) is what's achieved by raising \( b \) to the power of \( c \).
  • This conversion helps in visualizing the problem more easily and sets up the equation for solving.
In the given problem \( \log_{2} x = 5 \), it converts to \( 2^{5} = x \). This transformation lets us directly compute the value of \( x \) by calculating powers, which often feels more straightforward.
Solving Equations Efficiently
Solving equations requires understanding the path from the problem to the answer. After converting logarithmic equations to exponential form, the main task is to solve for the unknown. By computing the powers, you're essentially completing the equation.
  • For \( \log_{2} x = 5 \), we convert it to \( 2^{5} = x \).
  • Next, calculate \( 2^5 \). Multiplying, \( 2 \times 2 \times 2 \times 2 \times 2 \) gives us \( 32 \).
  • This shows that the value of \( x \) is \( 32 \).
Once you've solved for \( x \), you can check your work by substituting back. It confirms that you're correct if it holds true in the original equation. This method ensures clarity and confidence in solving similar problems.

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

For Problems \(21-30\), use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 0.008142\)

For Problems \(21-30\), use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 5\)

For Problems \(23-32\), approximate each of the following logarithms to three decimal places. $$ \log _{8} 514 $$

A piece of machinery valued at $$\$ 30,000$$ depreciates at a rate of \(10 \%\) yearly. How long will it take until it has a value of $$\$ 15,000$$ ?

For Problems \(21-30\), use your calculator to find each natural logarithm. Express answers to four decimal places. \(\ln 371.8\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.