/*! 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 25 Solve each exponential equation.... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 25

Solve each exponential equation. Express irrational solutions as decimals correct to the nearest thousandth. $$2(1.05)^{x}+3=10$$

Short Answer

Expert verified
The solution is approximately \( x \approx 25.673 \).

Step by step solution

01

Isolate the exponential term

First, subtract 3 from both sides of the equation to isolate the exponential term. \[ 2(1.05)^{x} + 3 - 3 = 10 - 3 \] This simplifies to: \[ 2(1.05)^{x} = 7 \]
02

Divide both sides by 2

Next, divide both sides of the equation by 2 to further isolate the exponential term. \[ \frac{2(1.05)^{x}}{2} = \frac{7}{2} \] This gives us: \[ (1.05)^{x} = 3.5 \]
03

Apply the natural logarithm

Now, take the natural logarithm of both sides to solve for \( x \). \[ \ln((1.05)^{x}) = \ln(3.5) \] Using the power rule of logarithms, this simplifies to: \[ x \cdot \ln(1.05) = \ln(3.5) \]
04

Solve for x

Now, solve for \( x \) by dividing both sides by \( \ln(1.05) \). \[ x = \frac{\ln(3.5)}{\ln(1.05)} \] Using a calculator, this evaluates to: \[ x \approx \frac{1.252763}{0.04879} \approx 25.673 \]

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Key Concepts

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

natural logarithms
Natural logarithms, denoted as \( \text{ln} \), are logarithms with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. They are particularly useful in solving exponential equations because they can simplify the exponents through their unique properties. When you take the natural logarithm of an exponential term, such as \( \text{ln}(a^b) \), you can use the power rule to bring the exponent down, making it easier to solve for variables like \( x \).
isolating exponential terms
To solve exponential equations, it's essential to isolate the exponential term. This means getting the exponential expression by itself on one side of the equation. Take the equation given in the exercise: 2(1.05)^{x} + 3 = 10. First, we subtract 3 from both sides to get 2(1.05)^{x} = 7. Next, divide both sides by 2 to isolate the exponential term, giving us (1.05)^{x} = 3.5. This process simplifies the equation and sets us up to apply logarithms.
irrational solutions
Irrational solutions are numbers that cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions. In this exercise, the solution is expressed irrationally as \( x = \frac{\text{ln}(3.5)}{\text{ln}(1.05)} \). This results in an irrational number, which we approximate to the nearest thousandth using a calculator, finding \( x \approx 25.673 \). It's important to understand that even though irrational numbers cannot be exactly written as a fraction or a terminating decimal, we can approximate them for practical use.
power rule of logarithms
The power rule of logarithms states that the logarithm of a number raised to a power can be rewritten as the power multiplied by the logarithm of the base: \( \text{log}(a^b) = b \cdot \text{log}(a) \). This rule is extremely helpful in solving equations involving exponents. In our exercise, we used this rule to simplify \( \text{ln}((1.05)^{x}) \) to \( x \cdot \text{ln}(1.05) \), making it easier to solve for \( x \). Essentially, the power rule helps to transform multiplicative relationships into additive ones, facilitating easier manipulation and solution of the equation.

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Most popular questions from this chapter

The amount of medication still available in the system is given by the function $$ f(t)=200(0.90)^{t} $$ In this model, \(t\) is in hours and \(f(t)\) is in milligrams. How long will it take for this initial dose to reach the dangerously low level of \(50 \mathrm{mg} ?\) Population Size Many environmental situations place effective limits on the growth of the number of an organism in an area. Many such limited-growth situations are described by the logistic function $$ G(x)=\frac{M G_{0}}{G_{0}+\left(M-G_{0}\right) e^{-k M x}} $$ where \(G_{0}\) is the initial number present, \(M\) is the maximum possible size of the population, and \(k\) is a positive constant. The screens illustrate a typical logistic function calculation and graph. (Graph can't copy) Assume that \(G_{0}=100, M=2500, k=0.0004,\) and \(x=\) time in decades ( \(10-\) yr periods). (a) Use a calculator to graph the function, using \(0 \leq x \leq 8,0 \leq y \leq 2500\) (b) Estimate the value of \(G(2)\) from the graph. Then evaluate \(G(2)\) algebraically to find the population after 20 yr. (c) Find the \(x\) -coordinate of the intersection of the curve with the horizontal line \(y=1000\) to estimate the number of decades required for the population to reach \(1000 .\) Then solve \(G(x)=1000\) algebraically to obtain the exact value of \(x .\)

Cost of Bread Assume the cost of a loaf of bread is \(\$ 4 .\) With continuous compounding, find the time it would take for the cost to triple at an annual inflation rate of \(4 \%\)

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-\sqrt{x^{2}-16}, \quad x \geq 4$$

Use the definition of inverses to determine whether \(f\) and \(g\) are inverses. $$f(x)=\frac{-1}{x+1}, \quad g(x)=\frac{1-x}{x}$$

World Population Growth since 2000 , world population in millions closely fits the exponential function $$ y=6084 e^{00120 x} $$ where \(x\) is the number of years since 2000 . (Source: U.S. Census Bureau.) (a) The world population was about 6853 million in 2010 . How closely does the function approximate this value? (b) Use this model to predict the population in 2020 . (c) Use this model to predict the population in 2030 . (d) Explain why this model may not be accurate for 2030 .
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