/*! 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 106 Solve \(\log _{8}(2 x+1)=-1\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 106

Solve \(\log _{8}(2 x+1)=-1\)

Short Answer

Expert verified
x = -\frac{7}{16}.

Step by step solution

01

Understand the logarithmic equation

A logarithmic equation of the form \(\text{log}_b(y) = x\) implies that \y = b^x\. Here, \(b = 8\), \y = 2x + 1\, and \x = -1\.
02

Rewrite the logarithmic form to exponential form

Rewrite the equation \(\text{log}_8(2x + 1) = -1\) to its exponential form: \(2x + 1 = 8^{-1}\).
03

Simplify the exponential term

Calculate \(8^{-1} \). Remember that negative exponents represent reciprocal values: \(8^{-1} = \frac{1}{8}\).
04

Set up the equation

Now, substitute \(8^{-1} = \frac{1}{8}\) back into the equation: \(2x + 1 = \frac{1}{8}\).
05

Solve for x

Isolate \x\ by subtracting 1 from both sides: \(2x = \frac{1}{8} - 1\). Convert 1 to a fraction with the same denominator: \(1 = \frac{8}{8}\). Thus, \(2x = \frac{1}{8} - \frac{8}{8} = \frac{1-8}{8} = \frac{-7}{8}\).
06

Finish solving for x

Divide both sides by 2 to isolate \x\: \( x = \frac{\frac{-7}{8}}{2} = \frac{-7}{16}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithms
Logarithms are mathematical operations that help us solve equations involving exponentiation. They answer the question: 'To what exponent must a base be raised to produce a given number?' For example, \(\text{log}_{10}(100) = 2\) because 10 raised to the power of 2 is 100.
exponential functions
Exponential functions involve terms in which a constant base is raised to a variable exponent. An example is \(b^x\). Exponential functions grow rapidly and are widely used in various fields such as finance and natural sciences. Converting between logarithmic and exponential forms is a key skill.
solving equations
Solving equations is the process of finding the variable values that make an equation true. It often involves multiple steps, such as isolating the variable, simplifying terms, and undoing operations. Practice and familiarity with different types of equations make this easier.
negative exponents
Negative exponents represent the reciprocal of the base raised to the corresponding positive exponent. For example, \(b^{-n} = \frac{1}{b^n}\). Understanding this concept is crucial for simplifying expressions and solving equations involving exponentiation.

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