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Chapter 4: Problem 6

Find the solution of the exponential equation, correct to four decimal places. $$3^{2 x-1}=5$$

Short Answer

Expert verified
The solution is approximately \(x = 1.2325\).

Step by step solution

01

Set Up the Equation

The given equation is \(3^{2x-1} = 5\). Our goal is to solve for \(x\).
02

Take the Natural Logarithm of Both Sides

To solve for \(x\), take the natural logarithm of both sides of the equation: \(\ln(3^{2x-1}) = \ln(5)\)
03

Use the Power Rule for Logarithms

Applying the power rule \(\ln(a^b) = b\ln(a)\), we can rewrite the left side: \((2x-1)\ln(3) = \ln(5)\)
04

Isolate the Term Involving x

First, divide both sides by \(\ln(3)\) to isolate the linear expression: \(2x-1 = \frac{\ln(5)}{\ln(3)}\).
05

Solve for x

Add 1 to both sides of the equation, then divide by 2 to solve for \(x\): \(2x = \frac{\ln(5)}{\ln(3)} + 1\)\(x = \frac{\left(\frac{\ln(5)}{\ln(3)} + 1\right)}{2}\)
06

Calculate the Value of x

Use a calculator to find \(x\). First calculate \(\frac{\ln(5)}{\ln(3)}\), and then apply the steps: \(\frac{\ln(5)}{\ln(3)} \approx 1.464974 \)Add 1 to this result:\(1.464974 + 1 = 2.464974\)Divide by 2:\(\frac{2.464974}{2} \approx 1.232487\) Rounding to four decimal places, \(x \approx 1.2325\).

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Key Concepts

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

solving logarithmic equations
Solving logarithmic equations involves using logarithms to find the unknown variable in an equation where the variable appears as an exponent. This technique is useful for equations where the standard algebraic methods don't easily apply due to the involvement of exponents. Here鈥檚 a simplified approach:
  • Take the natural logarithm or any logarithm of both sides of the equation to get the variable out of the exponent.
  • Use the properties of logarithms, such as the power rule, to simplify the expressions.
  • Isolate the variable using algebraic manipulations like addition, subtraction, multiplication, and division.
  • Perform calculations to find the approximate value of the variable.
By these methods, you can solve complex exponential equations with ease.
natural logarithm
The natural logarithm, denoted as \( \ln \)\, is a specific logarithm with base \(e\), where \(e\) is approximately equal to 2.71828. The natural logarithm is widely used in calculus and exponential equations because it simplifies the relationship between exponential functions and logarithms. Here鈥檚 why natural logarithms are important:
  • They directly relate to the exponential growth described by the Euler number, \(e\).
  • The rate of change in processes like compound interest and population growth can be effectively modeled using natural logarithms.
  • Natural logarithms have derivatives and integrals that are easy to compute, such as \(\frac{d}{dx}\ln(x) = \frac{1}{x} \).
The natural logarithm is a vital tool in solving exponential equations, transforming them into linear equations.
power rule for logarithms
The power rule for logarithms is a fundamental property that allows us to manipulate logarithmic expressions when they involve exponents. This rule states that \(\ln(a^b) = b\ln(a)\). By using the power rule, we can simplify expressions where the variable is in the exponent:
  • The exponent can be "brought down" in front of the logarithm, making it easier to solve for the variable.
  • This rule helps to linearize exponential equations, making them solvable using basic algebraic operations.
  • By applying this rule, you can greatly simplify solving equations such as \(3^{2x-1} = 5\) into a more manageable form.
Using the power rule reduces the complexity of logarithmic equations and is crucial in transforming exponential forms into their logarithmic counterparts.

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Most popular questions from this chapter

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Assume that a population of rabbits behaves according to the logistic growth model $$ n(t)=\frac{300}{0.05+\left(\frac{300}{n_{0}}-0.05\right) e^{-0.55 t}} $$ where \(n_{0}\) is the initial rabbit population. (a) If the initial population is 50 rabbits, what will the population be after 12 years? (b) Draw graphs of the function \(n(t)\) for \(n_{0}=50,500\) \(2000,8000,\) and \(12,000\) in the viewing rectangle \([0,15]\) by \([0,12,000]\) (c) From the graphs in part (b), observe that, regardless of the initial population, the rabbit population seems to approach a certain number as time goes on. What is that number? (This is the number of rabbits that the island can support.)
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