/*! 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 86 Solve each equation. If necessar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 86

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \log _{8}(2 x-1)=\frac{1}{3} $$

Short Answer

Expert verified
x = 1.5

Step by step solution

01

Write in exponential form

The given equation is \( \log _{8}(2 x-1)=\frac{1}{3} \). By converting it into exponential form, the equation becomes \( 8^{\frac{1}{3}} = 2 x - 1 \). This stems from the rule that \( \log _{b}(x) = y \) can be written as \( b^{y} = x \).
02

Calculate Base to the Power

Next is to calculate \( 8^{\frac{1}{3}} \). This operation represents the cube root of 8, which is equal to 2. Therefore, our equation now becomes \( 2 = 2 x - 1 \).
03

Solve for \( x \)

Finally, solve the equation \( 2 = 2 x - 1 \) for \( x \). Adding 1 to both sides gives \( 3 = 2x \). Dividing both sides by 2 gives \( x = 1.5 \).

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 Form
The exponential form of a logarithmic equation offers a different way to express the relationship between numbers. In our problem, we start by having the logarithmic equation \( \log _{8}(2 x-1)=\frac{1}{3} \).
This can be confusing if you're not familiar with how logs work. By converting this into exponential form, it becomes \( 8^{\frac{1}{3}} = 2x - 1 \).
So how do we go from a log equation to exponential form? It's actually easier than it sounds:
  • The base of the logarithm (which is 8 here) stays the base in the exponential form.
  • The result of the logarithmic function (\( \frac{1}{3} \)) becomes the exponent.
  • What's inside the log function (\( 2x-1 \)) becomes the expression that the base is equated to once raised to the power.
Remember, converting from logarithmic to exponential form is a method to solve the equation by switching perspectives. It allows us to work with exponential expressions, which might be more straightforward in certain contexts.
Cube Root
Taking roots, especially cube roots, is a fundamental concept in math. In our example, we calculate \( 8^{\frac{1}{3}} \). This expression asks for the cube root of 8. So, what does this mean?
Essentially, finding the cube root of a number is determining the value that, when multiplied by itself three times, gives the original number.
Let's break it down:
  • The cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).
  • When calculating roots, you are looking for a number that satisfies this multiplication condition.
  • You often see roots written as fractional exponents, like \( \frac{1}{3} \) indicates a cube root.
This step of finding the cube root transforms our exponential equation into something much simpler. With \( 8^{\frac{1}{3}} \) becoming the number 2, our equation is instantly more manageable: \( 2 = 2x - 1 \). This simplification is crucial for solving the equation.
Solving Equations
When solving equations, the goal is often to isolate the variable. This involves using a combination of algebraic techniques to find the value of \( x \). In our exercise, once simplified, our equation is \( 2 = 2x - 1 \). So how do we solve this?
Here's a step-by-step method for isolation:
  • First, add 1 to both sides of the equation to remove the constant term from the side with \( x \), leading to \( 3 = 2x \).
  • Second, divide both sides by 2, the coefficient of \( x \), to isolate \( x \).
  • This results in \( x = 1.5 \).
These steps ensure that the original relationship in the equation is maintained while manipulating it to find \( x \). Solving equations requires an understanding of maintaining balance鈥攚hatever you do to one side of the equation, you must also do to the other. This process guarantees that the solution is valid and accurately reflects the relationships within the problem.

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

Zoology Conservation efforts have increased the endangered Florida manatee population from 1465 in 1991 to 3276 in 2001 . If this growth rate continues, when might there be \(10,000\) manatees? Explain the reasoning behind your choice of a model.

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \log _{7} 3 x=3 $$

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 7^{x}-1=371 $$

Solve each equation. $$ \log 9-\log x+1=6 $$

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ \frac{1}{2} \log x+\log 4=2 $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

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.