/*! 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 46 Solve each logarithmic equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 46

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$1-4 \ln (2 x-1)=-5$$

Short Answer

Expert verified
\(x = \frac{e^{3/2} + 1}{2}\)

Step by step solution

01

Isolate the Logarithmic Term

To solve the logarithmic equation, we first need to isolate the logarithmic term. Starting from the equation: \[1 - 4 \ln(2x - 1) = -5\]Subtract 1 from both sides:\[-4 \ln(2x - 1) = -6\]
02

Divide Both Sides by -4

Now, divide both sides of the equation by -4 to solve for the natural logarithm:\[\ln(2x - 1) = \frac{6}{4} = \frac{3}{2}\]
03

Exponentiate to Eliminate the Logarithm

To eliminate the logarithm, exponentiate both sides using the base of the natural logarithm, which is \(e\). Hence, \[e^{\ln(2x-1)} = e^{\frac{3}{2}}\]This simplifies to:\[2x - 1 = e^{\frac{3}{2}}\]
04

Solve for x

Now solve for \(x\) by isolating it on one side of the equation:First, add 1 to both sides:\[2x = e^{\frac{3}{2}} + 1\]Then divide by 2:\[x = \frac{e^{\frac{3}{2}} + 1}{2}\]
05

Verify the Solution with a Calculator

Use a calculator to compute \(e^{\frac{3}{2}}\). If necessary, convert the result to a numerical value to verify the final solution. This checks that the solution satisfies the original equation as closely as possible, taking into account any rounding errors.

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.

Natural Logarithm
Natural logarithms are logs to the base of e, where e is approximately equal to 2.71828. The symbol for natural logarithm is \( \ln \). When you see \( \ln(x) \), it means "the power to which we must raise the number e to get x." Natural logarithms are particularly useful in calculus and can simplify complex multiplicative processes with their additive properties.
  • Given \( \ln(a) = b \), it indicates that \( e^b = a \).
  • In the logarithmic equation \( \ln(2x - 1) = \frac{3}{2} \), it tells us that the expression \( 2x - 1 \) can be rewritten in terms of the exponential function, namely: \( e^{\frac{3}{2}} \).
Understanding natural logarithms is crucial as they appear frequently in real-world situations involving exponential growth and decay, such as in population growth and radioactive decay.
Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself. For example, in \( 2^3 \), 2 is the base, and 3 is the exponent, meaning \( 2 \times 2 \times 2 \).
  • The property \( e^{\ln(x)} = x \) allows us to "undo" the logarithm. That's why in our step-by-step solution, we use exponentiation to solve \( \ln(2x - 1) = \frac{3}{2} \). Both sides of the equation are raised to the power of e.
  • Applying exponentiation helps remove the logarithm, simplifying the expression to \( 2x - 1 = e^{\frac{3}{2}} \).
Remember, exponentiation is the inverse operation of the logarithm, and this relationship is key to solving logarithmic equations.
Exact Form Solutions
In mathematics, an exact form solution provides the answer in terms of constants and variables without numerical approximations. Instead of converting \( e^{\frac{3}{2}} \) into a decimal, exact form means leaving it as \( e^{\frac{3}{2}} + 1 \). It provides exact values that are often more insightful, as any numerical approximation could introduce error.
  • In the logarithmic equation solution, we find \( x = \frac{e^{\frac{3}{2}} + 1}{2} \).
  • This expression is exact, relying on the fundamental constants like e, which is crucial for maintaining precise information in scientific and mathematical work.
Always aim for exact form solutions when possible, unless a numerical approximation is necessary for a practical decision or computation.
Use of Calculator in Algebra
Using a calculator in algebra can be invaluable when verifying results or converting exact expressions into decimal form. For example, when we solve for \( x \) in the equation and find \( x = \frac{e^{\frac{3}{2}} + 1}{2} \), using a calculator allows us to compute \( e^{\frac{3}{2}} \) to get a numerical approximation for practical interpretation.
  • Enter expressions carefully into the calculator to avoid errors. Ensure you're using parentheses correctly to maintain mathematical accuracy.
  • Use calculators to check your algebra by comparing the approximated decimal values against your expectations from the exact form solution.
Calculators are tools to aid in understanding, not replace fundamental algebraic expertise. Knowing when and how to use them enhances problem-solving efficiency.

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

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Find the present value of an account that will be worth \(\$ 10,000\) in 5 years, if interest is compounded semiannually at \(3 \%\).

A construction worker wants to invest \(\$ 60,000\) in a pension plan. One investment offers \(2 \%\) compounded quarterly. Another offers \(1.8 \%\) compounded continuously. Which investment will earn more interest in 5 years? How much more will the better plan earn?

Suppose that a sample of bacteria has a concentration of 2 million bacteria per milliliter and it doubles in concentration every 12 hours. Then the time \(T\) it takes for the sample to reach a concentration of \(C\) can be approximated by the following logarithmic function. \(T(C)=\frac{500}{29} \ln \frac{C}{2}\) (a) Find the domain of \(T .\) Interpret your answer. (b) How long does it take for the concentration of bacteria to increase by \(50 \% ?\) (c) Determine the concentration \(C\) after 15 hours by solving the equation \(T(C)=15\)

Newton's law of cooling says that the rate at which an object cools is proportional to the difference \(C\) in temperature between the object and the environment around it. The temperature \(f(t)\) of the object at time t in appropriate units after being introduced into an environment with a constant temperature \(T_{0}\) is $$f(t)=T_{0}+C e^{-k t}$$ where \(C\) and \(k\) are constants. Use this result. A piece of metal is heated to \(300^{\circ} \mathrm{C}\) and then placed in a cooling liquid at \(50^{\circ} \mathrm{C}\). After 4 minutes, the metal has cooled to \(175^{\circ} \mathrm{C}\). Estimate its temperature after 12 minutes.

Tree Growth The height of a tree in feet after \(x\) years is modeled by $$f(x)=\frac{50}{1+47.5 e^{-0.22 x}}$$ (a) Make a table for \(f\) starting at \(x=10\) and incrementing by \(10 .\) What seems to be the maximum height? (b) Graph \(f\) and identify the horizontal asymptote. Explain its significance. (c) After how long was the tree 30 feet tall?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

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.