/*! 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 100 Solve each equation. Find the ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 100

Solve each equation. Find the exact solutions. $$\log _{x}(9)=\frac{1}{2}$$

Short Answer

Expert verified
x = 81

Step by step solution

01

Rewrite the logarithmic equation in exponential form

Given the equation \(\text{log}_{x}(9) = \frac{1}{2}\), rewrite it in its exponential form. This translates to: \[x^{\frac{1}{2}} = 9.\]
02

Solve the exponential equation for x

To solve \(\text{x}^{\frac{1}{2}} = 9\), eliminate the exponent by squaring both sides of the equation: \[ \big(x^{\frac{1}{2}}\big)^2 = 9^2. \]This simplifies to: \[ x = 81. \]
03

Verify the solution

Substitute \x = 81\ back into the original logarithmic equation to ensure it's correct: \[ \text{log}_{81}(9) = \frac{1}{2}. \]

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
When solving logarithmic equations, the first step is often to rewrite the logarithm in its exponential form. This can make the equation easier to solve. In logarithmic terms, \(\text{log}_{x}(9) = \frac{1}{2}\) means 'x raised to what power gives 9?' To convert this into exponential form, we write: \[ x^{\frac{1}{2}} = 9. \] This transformation simplifies the problem by translating the logarithmic statement into one that's easier to manipulate and solve.
Logarithmic functions
Logarithms are the inverse functions of exponentials. This means they can reverse the process of exponential growth. For example, while exponentiation deals with raising a base to a power, logarithms help us find the exponent when the base and its power are known. Understanding this relationship is key to solving logarithmic equations like \(\text{log}_{x}(9) = \frac{1}{2}\). It tells us that if we know the logarithm, we can find the base or the power through the connection to its exponential form. The notation \(\text{log}_{b}(a) \) denotes the power to which the base \(\text{b}\) must be raised to yield \(\text{a}\).
Verifying solutions
After finding a potential solution, it's crucial to verify it by substituting it back into the original equation. This step confirms the accuracy of the solution. For our problem, we solved \(\text{x}^{\frac{1}{2}} = 9 \) to find that \(\text{x} = 81 \). To verify, we substitute \(\text{x} = 81 \) back into the original logarithmic equation: \[ \text{log}_{81}(9) = \frac{1}{2}. \]We check if this statement holds true. This verification process is vital as it ensures we have not made any errors during our calculations. Verification strengthens our confidence in the solution.

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

Visual Magnitude of a Star If all stars were at the same distance, it would be a simple matter to compare their brightness. However, the brightness that we see, the apparent visual magnitude \(m,\) depends on a star's intrinsic brightness, or absolute visual magnitude \(M_{V},\) and the distance \(d\) from the observer in parsecs ( 1 parsec \(=3.262\) light years), according to the formula \(m=M_{V}-5+5 \cdot \log (d) .\) The values of \(M_{V}\) range from \(-8\) for the intrinsically brightest stars to \(+15\) for the intrinsically faintest stars. The nearest star to the sun, Alpha Centauri, has an apparent visual magnitude of 0 and an absolute visual magnitude of 4.39 . Find the distance \(d\) in parsecs to Alpha Centauri.

Use the following definition. In chemistry, the \(\mathrm{pH}\) of a solution is defined to be $$\mathrm{pH}=-\log \left[H^{+}\right],$$ where \(H^{+}\) is the hydrogen ion concentration of the solution in moles per liter. Distilled water has a pH of approximately 7. A substance with a pH under 7 is called an acid, and one with a pH over 7 is called a base. The gastric juices in your stomach have a hydrogen ion concentration of \(10^{-1}\) moles per liter. Find the pH of your gastric juices.

Noise Pollution The level of a sound in decibels (db) is determined by the formula $$\text { sound level }=10 \cdot \log \left(I \times 10^{12}\right) \mathrm{db}$$ where \(I\) is the intensity of the sound in watts per square meter. To combat noise pollution, a city has an ordinance prohibiting sounds above 90 db on a city street. What value of \(I\) gives a sound of 90 db?

Marginal Revenue The revenue in dollars from the sale of \(x\) items is given by the function \(R(x)=500 \cdot \log (x+1) .\) The marginal revenue function \(M R(x)\) is the difference quotient for \(R(x)\) when \(h=1 .\) Find \(M R(x)\) and write it as a single logarithm. What happens to the marginal revenue as \(x\) gets larger and larger?

Economic Impact An economist estimates that \(75 \%\) of the money spent in Chattanooga is respent in four days on the average in Chattanooga. So if \(P\) dollars are spent, then the economic impact \(I\) in dollars after \(n\) respendings is given by $$I=P(0.75)^{n}$$ When \(I <0.02 P,\) then \(P\) no longer has an impact on the economy. If the Telephone Psychics Convention brings \(\$ 1.3\) million to the city, then in how many days will that money no longer have an impact on the city?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.