/*! 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 71 Find each value of x. $$ \lo... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 71

Find each value of x. $$ \log _{8} x=2 $$

Short Answer

Expert verified
The value of \(x\) is 64.

Step by step solution

01

Understand the Logarithmic Equation

You are given the logarithmic equation \(\log_{8} x = 2\). This means that \(8\) raised to some power equals \(x\). In this equation, the base is \(8\) and it solves for \(x\).
02

Convert Logarithmic to Exponential Form

To solve for \(x\), we need to convert the logarithmic equation into exponential form. The logarithmic equation \(\log_{8} x = 2\) can be rewritten in exponential form as \(8^2 = x\).
03

Calculate the Exponential

Now that you have the equation \(8^2 = x\), calculate \(8^2\). Multiply 8 by itself to get \(8 \times 8 = 64\). Thus, \(x = 64\).

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Key Concepts

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

Exponential Form
When dealing with logarithmic equations, understanding how to convert them into exponential form is crucial. A logarithm answers the question: "To what power must the base be raised, to get this number?" For example, in the equation \( \log_{8} x = 2 \), we ask: "8 raised to what power equals \(x\)?" The answer is the exponential form of the equation.To change a logarithm into its exponential form:
  • The base of the logarithm becomes the base of the exponential expression.
  • The answer to the logarithm is the exponent to which the base is raised.
  • The result of the exponential raised power equals the value inside the log function.
Thus, \( \log_{8} x = 2 \) transforms into the exponential equation \( 8^2 = x \). This conversion clarifies the relationship between the components of the logarithm.
Base of a Logarithm
The base of a logarithm is a fundamental aspect that significantly determines the outcome of the equation. It is the number that we repeatedly multiply by itself, to reach the number inside the logarithm.In \( \log_{8} x = 2 \), the base is 8. This means we are dealing with powers of 8.Here's how to think about the base in simple terms:
  • The base tells us what number is being used for repeated multiplication.
  • Changing the base changes the entire equation and its result.
The base helps transition the problem into a more workable form, by helping us shift our focus from a logarithmic to an exponential perspective. By identifying the base, we can solve for \(x\) by performing exponentiation, as shown in the original problem.
Solving for x
The ultimate goal is to find the value of \(x\). We do this by applying the information gleaned from the logarithmic form and the exponential conversion.From the problem, we have the equation \( \log_{8} x = 2 \). By converting to exponential form, we get \( 8^2 = x \).Now comes the calculation:
  • Recognize that \(8^2\) means multiply 8 by itself: \(8 \times 8\).
  • This works out to \(64\), giving us \(x = 64\).
Solving for \(x\) means carrying out the multiplication to find a precise numerical answer, using the exponential equation derived from the original logarithmic statement. This step-by-step process allows you to solve similar logarithmic equations confidently.

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