/*! 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 66 Solve equation. \(\log _{x} 64... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 66

Solve equation. \(\log _{x} 64=2\)

Short Answer

Expert verified
The solution is \(x = 8\).

Step by step solution

01

- Understand the logarithmic equation

The equation given is \(\re \log_{x} 64 = 2 \). This means that the base \(x\) raised to the power of 2 equals 64.
02

- Convert the logarithmic equation to an exponential form

Rewrite the equation \(\re \log_{x} 64 = 2 \) in its exponential form. This gives: \(\re x^{2} = 64 \).
03

- Solve for the base \(x\)

To solve for \(x\), take the square root of both sides. This gives: \(\re \sqrt{x^2} = \sqrt{64} \). Therefore, \(x = \pm8\).
04

- Determine the valid solution

Since logarithms are generally defined for positive bases, \(x\) must be a positive number. Hence, the valid base \(x\) is 8.

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Key Concepts

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

exponential form
In mathematical terms, the relationship between logarithms and exponentials is foundational. When you have an equation like \(\re \log_{x} 64 = 2 \), it tells you something specific in exponential terms. The logarithmic equation \(\re \log_{x} 64 = 2 \) can be rephrased as an exponential equation. This means the base \(\re x \) raised to the power of 2 equals 64. When converting from logarithmic to exponential form, you follow this pattern: if \(\re \log_{b}(a) = c \), then \(\re b^{c} = a \).

So, from \(\re \log_{x} 64 = 2 \), you write \(\re x^{2} = 64 \). This conversion makes it easier to solve because you might find exponential equations more familiar or intuitive.
solving equations
Once you have the exponential form \(\re x^{2} = 64 \), solving it requires finding the value of \(\re x \) that makes the equation true. To solve the equation \(\re x^{2} = 64 \), you need to take the square root of both sides. The square root operation helps to 'undo' the exponentiation. This means: \(\re \sqrt{x^{2}} = \sqrt{64} \).

The solution to \(\re x^{2} = 64 \) is \(\re x = \pm 8 \). Both \(\re 8 \) and \(\re -8 \) are valid solutions in mathematical terms because squaring either 8 or -8 results in 64.
positive bases
Logarithms have specific rules, one of which is that the base must be a positive number. Therefore, even though \(\re x = \pm 8 \) solves the exponential equation, only the positive base makes sense for the original logarithmic form.

This is because logarithms are not defined for negative bases. So in the context of logarithms, the valid solution here is \(\re x = 8 \).

The reasoning follows this rule: Logarithms can only operate with positive numbers as their bases, which ensures that they follow their natural properties and avoid ambiguities. Always remember to verify solutions to see if they make sense in the context of the problem you are solving.

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