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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.42
Chapter 1, Problem 1.42

Solve each equation.
48=6e4k48=6e^{4k}

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1
Start by isolating the exponential term. Divide both sides of the equation by 6 to get: \( e^{4k} = \frac{48}{6} \).
Simplify the right side of the equation: \( e^{4k} = 8 \).
To solve for \( k \), take the natural logarithm of both sides: \( \ln(e^{4k}) = \ln(8) \).
Use the property of logarithms that \( \ln(e^x) = x \) to simplify the left side: \( 4k = \ln(8) \).
Finally, solve for \( k \) by dividing both sides by 4: \( k = \frac{\ln(8)}{4} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. These functions model growth or decay processes and are characterized by their rapid increase or decrease. Understanding how to manipulate and solve equations involving exponential functions is crucial for solving problems like the one presented.
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Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form a^x = b. The natural logarithm, denoted as ln, specifically deals with the base 'e'. In the context of the given equation, applying logarithms helps isolate the variable 'k' by transforming the exponential equation into a linear form, making it easier to solve.
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Logarithms Introduction

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables or solve for unknowns. This includes operations such as addition, subtraction, multiplication, division, and applying properties of equality. In the context of the given equation, effective algebraic manipulation is necessary to isolate 'e^(4k)' and subsequently apply logarithmic properties to find the value of 'k'.
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Related Practice
Textbook Question

Convert the following expressions to the indicated base.


a1lnaa^{\(\frac{1}{\ln a}\)} using basa e, for a>0a > 0 and a≠1a ≠ 1

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Textbook Question

Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator.


cos⁻¹ √3/2

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Textbook Question

Working with composite functions Find possible choices for outer and inner functions ƒ and g such that the given function  h equals ƒ o g .


h(x) = √ (x⁴ + 2 )

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Textbook Question

Working with composite functions

Find possible choices for outer and inner functions ƒ and g such that the given function h equals ƒ o g.

h(x) = (2) / ( x⁶ + x² + 1)²

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Textbook Question

Find the inverse f−1(x)f^{-1}\(\left\)(x\(\right\)) of each function (on the given interval, if specified).

f(x)=e2x+6f\(\left\)(x\(\right\))=e^{2x+6}

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Textbook Question

Solving equations Solve each equation.

√2 sin 3Θ + 1 = 2, 0 ≤ Θ ≤ π

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