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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 1.3.59
Chapter 1, Problem 1.3.59

Solving equations Solve the following equations.


3(ˣ³⁻⁴) = 15

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1
Step 1: Start by isolating the exponential expression. Divide both sides of the equation by 3 to simplify: \( \frac{3(x^3 - 4)}{3} = \frac{15}{3} \).
Step 2: Simplify the equation from Step 1. This results in \( x^3 - 4 = 5 \).
Step 3: Solve for \( x^3 \) by adding 4 to both sides of the equation: \( x^3 = 5 + 4 \).
Step 4: Simplify the right side of the equation from Step 3: \( x^3 = 9 \).
Step 5: Solve for \( x \) by taking the cube root of both sides: \( x = \sqrt[3]{9} \).

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Key Concepts

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

Exponential Equations

Exponential equations involve variables in the exponent, such as x in the expression a^x. To solve these equations, one often uses logarithms to isolate the variable. Understanding the properties of exponents and logarithms is crucial for manipulating and solving these types of equations.
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Solving Exponential Equations Using Logs

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable on one side and all other terms on the opposite side. In the given equation, this means simplifying and dividing to find the value of x.
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Properties of Equality

The properties of equality state that if two expressions are equal, then one can be manipulated without changing the equality. This includes adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number. These properties are essential for solving equations systematically.
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Properties of Functions
Related Practice
Textbook Question

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

f(x)=2x2+1f\(\left\)(x\(\right\))=\(\frac{2}{x^2+1}\), for x≥0x\(\geq{0}\)

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Textbook Question

Composite functions and notation

Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).

Simplify or evaluate the following expressions.

g(ƒ(u))

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Textbook Question

Solving equations Solve the following equations.


log₁₀ x= 3

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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)=xx−2f\(\left\)(x\(\right\))=\(\frac{x}{x-2}\), for x>2x\(\gt{2}\)

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Textbook Question

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.


logbx²

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Textbook Question

Solving equations Solve each equation.


sin² 2Θ = 1/2, -π/2 ≤ Θ ≤ π/2

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