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Chapter 1: Problem 47

Solve each equation for \(x\) . $$(a)e^{7-4 x}=6 \quad \text { (b) } \ln (3 x-10)=2$$

Short Answer

Expert verified
For (a), \( x = \frac{7-\ln(6)}{4} \). For (b), \( x = \frac{e^2 + 10}{3} \).

Step by step solution

01

Solving Part (a)

To solve the equation \( e^{7-4x} = 6 \), we start by taking the natural logarithm on both sides to remove the exponential. This gives us: \( \ln(e^{7-4x}) = \ln(6) \).
02

Apply Logarithm Properties

Using the logarithmic property \( \ln(e^y) = y \), we simplify the left side: \( 7-4x = \ln(6) \). Now the equation becomes: \( 7 - 4x = \ln(6) \).
03

Isolate x

To isolate \( x \), first subtract 7 from both sides: \( -4x = \ln(6) - 7 \). Then, divide by \(-4\) to solve for \( x \): \( x = \frac{7-\ln(6)}{4} \).
04

Solving Part (b)

For the equation \( \ln(3x-10) = 2 \), exponentiate both sides to remove the logarithm: \( e^{\ln(3x-10)} = e^2 \). This simplifies to \( 3x - 10 = e^2 \).
05

Solve for x

Add 10 to both sides to isolate the \( 3x \) term: \( 3x = e^2 + 10 \). Then divide both sides by 3 to solve for \( x \): \( x = \frac{e^2 + 10}{3} \).

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a special kind of logarithm that uses the constant \( e \) (approximately 2.71828) as its base. This type of logarithm helps us solve equations involving exponential growth or decay. The natural logarithm has useful properties:
  • \( \ln(1) = 0 \)
  • \( \ln(e) = 1 \)
  • \( \ln(e^y) = y \)
Using these properties makes it straightforward to work with equations involving the natural exponent \( e \). For example, if you need to solve an equation like \( \ln(y) = x \), you can convert it to exponential form: \( y = e^x \). Natural logarithms are integral in calculus and various real-world phenomena in biology, economics, and physics.
Exponential Functions
Exponential functions take the form \( f(x) = a \times e^{bx} \), where:
  • \( a \) is a constant multiplier
  • \( e \) is the base of the natural logarithm
  • \( b \) is the rate of growth or decay
  • \( x \) is the exponent variable
These functions model many natural processes, such as population growth, radioactive decay, and interest calculations. The key feature of exponential functions is their rapid increase or decrease. In equation solving, if you want to eliminate the exponential function, you use the natural logarithm. For example, given \( e^{7 - 4x} = 6 \), applying \( \ln \) to both sides helps in simplifying to a linear form manageable to solve for \( x \). Thus, understanding exponential functions is crucial to solving these types of problems.
Solving Equations
Solving equations, especially involving exponential functions and logarithms, often requires a methodical approach. Let's break it down:
  • Identify the type of equation you're dealing with 鈥� e.g., exponential, logarithmic.
  • Choose the correct strategy to isolate the variable - either by applying logarithms or exponentials as shown in the solution steps.
  • Simplify the equation by using logarithm properties or algebraic manipulations.
For instance, when you have \( e^{7-4x} = 6 \):- Taking \( \ln \) of both sides removes the exponent.- Clear up and isolate \( x \) using subtraction and division.Similarly, when solving the equation \( \ln(3x-10) = 2 \):- Raise \( e \) to the power of both sides to eliminate \( \ln \).- Solve the resulting linear equation for \( x \).Approaching equations systematically, particularly using properties of \( \ln \) and exponential functions, streamlines the problem-solving process.

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Most popular questions from this chapter

An isotope of sodium, 24 \(\mathrm{Na}\) , has a half-life of 15 hours. A sample of this isotope has mass 2 \(\mathrm{g}\) . (a) Find the amount remaining after 60 hours. (b) Find the amount remaining after \(t\) hours. (c) Estimate the amount remaining after 4 days. (d) Use a graph to estimate the time required for the mass to be reduced to 0.01 \(\mathrm{g} .\)

(a) How is the logarithmic function \(y=\log _{b} x\) defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function \(y=\log _{b} x\) if \(b>1\)

Find \(f \circ g \circ h\) \(f(x)=\tan x, \quad g(x)=\frac{x}{x-1}, \quad h(x)=\sqrt[1]{x}\)

Find the exact value of each expression. (a) \(\log _{5} 125\) (b) \(\log _{3}\left(\frac{1}{27}\right)\)

Find the exact value of each expression. (a) $$e^{-2 \ln 5} \quad (b) \ln \left(\ln e^{e^{\prime \prime}}\right)$$
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