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Chapter 4: Problem 21

Find the solution of the exponential equation, rounded to four decimal places. $$5^{x}=4^{x+1}$$

Short Answer

Expert verified
The solution is approximately 6.2224.

Step by step solution

01

Introduce Logarithms

Start by applying the logarithm to both sides of the equation to use the property that allows exponents to be brought down as coefficients. The equation becomes: \[\log(5^{x}) = \log(4^{x+1})\]
02

Apply Logarithmic Power Rule

Use the power rule of logarithms, which states \(\log(a^{b}) = b \cdot \log(a)\), to rewrite both sides:\[x \cdot \log(5) = (x+1) \cdot \log(4)\]
03

Expand and Rearrange Equation

Expand the right side of the equation:\[x \cdot \log(5) = x \cdot \log(4) + \log(4)\]Then, rearrange the terms to get all terms involving \(x\) on one side:\[x \cdot \log(5) - x \cdot \log(4) = \log(4)\]
04

Factor Out x

Factor \(x\) out from the left side of the equation:\[x(\log(5) - \log(4)) = \log(4)\]
05

Solve for x

Divide both sides by \((\log(5) - \log(4))\) to isolate \(x\):\[x = \frac{\log(4)}{\log(5) - \log(4)}\]
06

Calculate x

Use a calculator to compute the logarithms and solve for \(x\). First, calculate \(\log(4)\), \(\log(5)\). Then, substitute these values into the equation:\[x = \frac{\log(4)}{\log(5) - \log(4)} \approx \frac{0.6021}{0.6989 - 0.6021} \approx \frac{0.6021}{0.0968} \approx 6.2224\]
07

Verify and Round the Solution

Verify by substituting back into the original equation to check that both sides are approximately equal. If they are, then round to four decimal places. The solution is already rounded to four decimal places as: 6.2224.

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Key Concepts

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

Logarithms
Logarithms are a mathematical tool used to solve equations involving exponents, especially when variables are in the exponent position. They allow us to transform multiplicative problems into additive ones, simplifying the process. In the equation \(5^x = 4^{x+1}\), applying logarithms to both sides helps us use properties of logarithms to make solving for \(x\) easier.

Key properties of logarithms include:
  • The logarithm of a product: \(\log(a \cdot b) = \log(a) + \log(b)\)
  • The logarithm of a quotient: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
  • The logarithm of a power: \(\log(a^b) = b \cdot \log(a)\)
Applying these properties can break down complex exponential expressions into manageable linear equations. By introducing logarithms to both sides of an exponential equation, we create a path to isolate the variable and solve the equation.
Logarithmic Power Rule
The logarithmic power rule is a crucial property that simplifies working with exponents in logarithmic expressions. It states that for any positive number \(a\) and real number \(b\), \(\log(a^b) = b \cdot \log(a)\). This rule allows us to "bring down" the exponent in a logarithmic expression, turning it into a multiplication problem instead.

In our context, for the equation \(\log(5^x) = \log(4^{x+1})\), by applying the power rule, we transform the equation into \(x \cdot \log(5) = (x+1) \cdot \log(4)\).
  • Both expressions are simplified by bringing the exponents in front of the logarithms.
  • This results in an equation where \(x\) is part of multiplying terms rather than an exponent, simplifying isolation of \(x\).
The logarithmic power rule is essential for tackling exponential equations where isolating the variable in the exponent is necessary for solving the equation.
Solving Equations
Solving equations involves a series of steps aimed at isolating the variable of interest. For exponential equations, this often requires transforming them into linear equations using logarithms. Here's an approach demonstrated in solving \(5^x = 4^{x+1}\):

  • Apply logarithms to both sides to make use of logarithmic properties.
  • Employ the logarithmic power rule to manage the exponents.
  • Rearrange terms to collect like terms on one side of the equation. For example, \(x \cdot \log(5) = x \cdot \log(4) + \log(4)\) becomes \(x \cdot \log(5) - x \cdot \log(4) = \log(4)\).
  • Factor out the variable \(x\) from one side. This makes the left hand side of the equation a product of \(x\) and a constant coefficient.
  • Divide both sides by this coefficient to solve for \(x\). In this problem, \(x = \frac{\log(4)}{\log(5) - \log(4)}\).
This structured process leads to finding an exact or approximate solution. Calculators are typically required for calculating logarithms to find the numerical solution. Verifying a solution includes substituting it back into the original equation, ensuring both sides are nearly equal, confirming the result. When required, as in this exercise, the solution can be rounded to a specified precision, such as four decimal places.

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Most popular questions from this chapter

Fish Population \(A\) small lake is stocked with a certain species of fish. The fish population is modeled by the function $$P=\frac{10}{1+4 e^{-0.8 t}}$$ where \(P\) is the number of fish in thousands and \(t\) is measured in years since the lake was stocked. (a) Find the fish population after 3 years. (b) After how many years will the fish population reach 5000 fish?

Compound Interest A woman invests \(\$ 6500\) in an account that pays \(6 \%\) interest per year, compounded continuously. (a) What is the amount after 2 years? (b) How long will it take for the amount to be \(\$ 8000 ?\)

Vilfredo Pareto \((1848-1923)\) observed that most of the wealth of a country is owned by a few members of the population. Pareto's Principle is $$\log P=\log c-k \log W$$ where \(W\) is the wealth level (how much money a person has) and \(P\) is the number of people in the population having that much money. (a) Solve the equation for \(P\). (b) Assume that \(k=2.1, c=8000,\) and \(W\) is measured in millions of dollars. Use part (a) to find the number of people who have \(\$ 2\) million or more. How many people have \(\$ 10\) million or more?

Discuss each equation and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which any term is undefined.) (a) \(\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}\) (b) \(\log _{2}(x-y)=\log _{2} x-\log _{2} y\) (c) \(\log _{5}\left(\frac{a}{b^{2}}\right)=\log _{5} a-2 \log _{5} b\) (d) \(\log 2^{2}=z \log 2\) (e) \((\log P)(\log Q)=\log P+\log Q\) (f) \(\frac{\log a}{\log b}=\log a-\log b\) (g) \(\left(\log _{2} 7\right)^{x}=x \log _{2} 7\) (h) \(\log _{a} a^{a}=a\) (i) \(\log (x-y)=\frac{\log x}{\log y}\) (j) \(-\ln \left(\frac{1}{A}\right)=\ln A\)

Draw the graph of the function in a suitable viewing rectangle, and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$y=\frac{\ln x}{x}$$
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