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Chapter 4: Problem 54

Solve each equation. Round approximate solutions to four decimal places. $$500(1.06)^{x}=400(1.02)^{4 x}$$

Short Answer

Expert verified
x \approx 10.6770

Step by step solution

01

Rewrite the equations in standard form

First, we need to isolate the exponential expressions. The given equation is:500(1.06)^x = 400(1.02)^{4x}Divide both sides by 400:\(\frac{500}{400}(1.06)^x = (1.02)^{4x}\)Which simplifies to:\(1.25(1.06)^x = (1.02)^{4x}\)
02

Take logarithms of both sides

To solve for \(x\), take the natural logarithm (ln) of both sides to get:\(\ln[1.25(1.06)^x] = \ln[(1.02)^{4x}]\)
03

Use logarithm properties

Apply the properties of logarithms to separate the terms. The property \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(a^b) = b \cdot \ln(a)\) will be used here:\(\ln(1.25) + \ln((1.06)^x) = \ln((1.02)^{4x})\)Then:\(\ln(1.25) + x \cdot \ln(1.06) = 4x \cdot \ln(1.02)\)
04

Isolate the variable x

Rearrange the equation to solve for \(x\):\(\ln(1.25) = 4x \cdot \ln(1.02) - x \cdot \ln(1.06)\)Factor out \(x\) on the right-hand side:\(\ln(1.25) = x(4 \cdot \ln(1.02) - \ln(1.06))\)Divide both sides by \(4 \cdot \ln(1.02) - \ln(1.06)\) to solve for \(x\):\(x = \frac{\ln(1.25)}{4 \cdot \ln(1.02) - \ln(1.06)}\)
05

Calculate the value of x

Now calculate the value of \(x\) using a calculator. Note that \(\ln(1.25) \approx 0.2231\), \(\ln(1.02) \approx 0.0198\), and \(\ln(1.06) \approx 0.0583\)\(x \approx \frac{0.2231}{4 \cdot 0.0198 - 0.0583} \approx \frac{0.2231}{0.0792 - 0.0583} \approx \frac{0.2231}{0.0209} \approx 10.6770\)

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Key Concepts

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

Logarithms
Logarithms are a way to express exponentiation in reverse. In simplest terms, if you have a number that is the result of raising a base to a certain power, the logarithm gives you the exponent.
For example, if you have \(b^y = x\), then the logarithm of \(x\) with base \(b\) is \(y\). This is written as \(\log_b(x) = y\).
Logarithms can be very useful when solving exponential equations since they allow you to work with the exponents directly.
In most calculations, we use either the common logarithm (\(\log\), with base 10) or the natural logarithm (\(\ln\), with base \(e\)).
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a logarithm with base \(e\), where \(e\) is an irrational number approximately equal to 2.71828.
The natural logarithm is particularly useful in many areas of mathematics and science because it is the inverse function of the exponential function \(e^x\).
For example, if \(e^y = x\), then \(\ln(x) = y\).
This makes it handy for solving equations where the variable is in the exponent.
In the example equation \(500(1.06)^x = 400(1.02)^{4x}\), we used the natural logarithm to bring down the exponents and solve for \(x\).
Properties of Logarithms
Logarithms have several useful properties that make them invaluable for solving exponential equations:
1. **Product Property**: \(\ln(ab) = \ln(a) + \ln(b)\). This property allows you to split the logarithm of a product into the sum of logarithms.
2. **Quotient Property**: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\). This property lets you break down the logarithm of a quotient into the difference of logarithms.
3. **Power Property**: \(\ln(a^b) = b \cdot \ln(a)\). This lets you bring down an exponent so it can be manipulated more easily.
These properties were used to simplify the equation and isolate the variable. For example, we used the power property to extract \(x\) in the equation \(\ln((1.06)^x)\).
Solving Exponential Equations
Solving exponential equations often involves the following steps:
1. **Isolate the Exponentials**: Try to get the exponential terms by themselves on one side of the equation.
2. **Apply Logarithms**: Take the natural logarithm (or common logarithm) of both sides to make the exponent a multiplicative term.
3. **Use Logarithm Properties**: Apply properties of logarithms to simplify and solve for the variable.
4. **Solve for the Variable**: Isolate the variable and solve using basic algebra.
For instance, in the problem \(500(1.06)^x = 400(1.02)^{4x}\), we isolated the exponential terms, applied the natural logarithm, utilized logarithm properties to separate the terms, and finally solved for \(x\).
This systematic approach simplifies the process of dealing with exponential equations.

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Most popular questions from this chapter

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Rewrite each expression as a sum or difference of logarithms. $$\log _{3}(x y)$$
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