Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Exponential & Logarithmic Equations

Calculus

0. Functions

Exponential & Logarithmic Equations

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1:23 minutes

Problem 1.51

Textbook Question

Solving equations Solve the following equations.

log₁₀ x= 3

Verified step by step guidance

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.

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.

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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Textbook Question

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} ƒ(x) = 1/(x-5)