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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 1.51
Chapter 1, Problem 1.51

Solving equations Solve the following equations.


log₁₀ x= 3

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1
Recognize that the equation \( \log_{10} x = 3 \) is in logarithmic form.
Recall the definition of a logarithm: \( \log_b a = c \) means \( b^c = a \).
Apply the definition to the given equation: \( \log_{10} x = 3 \) implies \( 10^3 = x \).
Calculate \( 10^3 \) to find the value of \( x \).
Conclude that \( x \) is the result of \( 10^3 \).

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Key Concepts

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

Logarithms

Logarithms are the inverse operations of exponentiation. The equation log₁₀ x = 3 means that 10 raised to the power of 3 equals x. Understanding logarithms is essential for solving equations involving them, as they help to express relationships between numbers in a more manageable form.
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Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the context of the logarithmic equation, recognizing that the logarithm represents an exponent allows us to convert the logarithmic form into an exponential form, facilitating the solution process.
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Properties of Logarithms

Properties of logarithms, such as the product, quotient, and power rules, provide tools for simplifying and manipulating logarithmic expressions. While not directly needed for this specific equation, understanding these properties can be crucial for solving more complex logarithmic equations and for combining multiple logarithmic terms.
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Related Practice
Textbook Question

Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

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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)=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.


3(ˣ³⁻⁴) = 15

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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

Solving equations Solve each equation.


sin² 2Θ = 1/2, -π/2 ≤ Θ ≤ π/2

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