/*! 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 49 Solve. See Example 4. $$ \lo... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 49

Solve. See Example 4. $$ \log _{x} 49=2 $$

Short Answer

Expert verified
The solution is \( x = 7 \).

Step by step solution

01

Understanding the Equation

The equation given is \( \log_x 49 = 2 \). This means that \( x \) is the base of the logarithm that results in 49 when raised to the power of 2.
02

Converting Logarithmic Equation to Exponential Form

Rewrite the logarithmic equation \( \log_x 49 = 2 \) in its exponential form. This gives us the equation \( x^2 = 49 \).
03

Solving the Exponential Equation

To find \( x \), solve the equation \( x^2 = 49 \). Taking the square root of both sides, we have two potential solutions: \( x = 7 \) and \( x = -7 \). However, since a logarithm's base must be positive, only \( x = 7 \) is valid.

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Key Concepts

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

Exponential Form
When you come across a logarithmic equation like \( \log_x 49 = 2 \), you are essentially dealing with a way to express an equation involving exponents. A logarithm answers the question: "To what power must the base be raised, to produce a given number?" In this case, we convert the logarithmic form to an exponential form to make it easier to solve. This is achieved by rewriting:
  • The base \( x \)
  • The power it is raised to \( 2 \)
  • The result \( 49 \)
This conversion allows you to see the problem as \( x^2 = 49 \), a simple equation to work with.Understanding exponential form greatly simplifies interpretations of logarithmic equations, as it provides a direct view of how numbers relate via powers.
Solving Logarithmic Equations
To solve logarithmic equations, the key is to transform them into equations that are easier to work with鈥攐ften exponential form. Our equation \( \log_x 49 = 2 \) transforms into \( x^2 = 49 \). Here鈥檚 how to proceed:
  • Transform the logarithmic equation into an exponential equation.
  • Upon getting \( x^2 = 49 \), solve for the unknown \( x \).
  • Take the square root of both sides: \( x = \pm7 \).
Yet, in the context of a logarithmic base, \( x \) must be a positive number. Thus, we discard \( x = -7 \) and keep \( x = 7 \) as the solution.Solving logarithmic equations requires understanding both logarithmic concepts and the conventions surrounding exponents, especially because logarithm bases must be positive.
Properties of Logarithms
Logarithms follow certain properties that make solving equations and understanding transformations more manageable. Understanding these properties can make dealing with complex equations more intuitive. Important properties include:
  • Product Property: \( \log_b (MN) = \log_b M + \log_b N \)
  • Quotient Property: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
  • Power Property: \( \log_b (M^k) = k \cdot \log_b M \)
For our example equation \( \log_x 49 = 2 \), the concept of expressing this in exponential form is leveraged through the understanding that \( \log_x x^2 \) would give 2, reinforcing the practical utility of logarithmic properties in solving equations. These properties are fundamental to progressing in mathematical problem-solving, helping identify, separate, and simplify components of logarithmic expressions.

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Most popular questions from this chapter

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\log (x+2) $$

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. Graph \(f(x)=\ln x \text { (Exercise } 89), f(x)=\ln x-3\) (Exercise \(95), \text { and } f(x)=\ln x+3 \text { (Exercise } 96)\) on the same screen. Discuss any trends shown on the graphs.

Use a graphing calculator to solve each equation. For example, to solve Exercise \(73,\) let \(Y_{1}=e^{0.3 x}\) and \(Y_{2}=8\) and graph the equations The \(x\) -value of the point of intersection is the solution. Round all solutions to two decimal places. $$ e^{0.3 x}=8 $$

Graph each function and its inverse function on the same set of axes. Label any intercepts. $$ y=\left(\frac{1}{3}\right)^{x} ; y=\log _{1 / 3} x $$

Simplify. See Example 5. $$ \log _{2} 2 $$
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