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Chapter 12: Problem 9

Write in logarithmic form. \(10^{-3}=0.001\)

Short Answer

Expert verified
\[ \log_{10}(0.001) = -3 \]

Step by step solution

01

Identify the Components

In the equation given, identify the base, the exponent, and the result. Here, the base is 10, the exponent is -3, and the result is 0.001.
02

Understand the Logarithmic Form

The logarithmic form of an equation expresses the exponent as a logarithm. The general form is \[a^b = c \] rewritten as \[ \log_a (c) = b \].
03

Rewrite the Equation

Using the components identified, rewrite the given exponential equation in logarithmic form: \[10^{-3} = 0.001 \] becomes \[ \log_{10} (0.001) = -3 \].

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Key Concepts

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

Exponential Equations
An exponential equation is a mathematical statement where an expression is set equal to a power of a base. It's of the form y = a^x where 'a' is the base, 'x' is the exponent, and 'y' represents the result. In practical terms, exponential equations are useful for modeling growth and decay, such as population growth or radioactive decay. Understanding how to convert these to logarithmic form can greatly simplify solving them.
  • Base: The constant value that is raised to a power.
  • Exponent: The power to which the base is raised.
  • Result: The outcome of the base raised to the exponent.
The conversion from logarithmic to exponential form essentially allows us to handle equations involving very large or small numbers more easily.
Logarithms
Logarithms are the inverse operation to exponentiation. They allow us to determine the exponent that a base must be raised to yield a given number. For example, if we have 10^2 = 100 then log_{10}(100) = 2. The main aim of logarithms is to uncover the exponent in an equation. They are immensely useful across different scientific and engineering fields.
Converting an exponential equation to logarithmic form can provide several benefits:
  • Makes solving complex calculations more manageable.
  • Transforms multiplicative processes into additive ones.
  • Helps in solving equations with unknown exponents.
The general form of a logarithm is x = log_a(y) which reads as 'x is the exponent to which the base a must be raised to yield y'.
Base of a Logarithm
The base of a logarithm is the number that is raised to some power to produce a given number. In the equation log_a(b) = c, 'a' is the base. The base is a critical component, as changing the base alters the value of the logarithm. Common bases include 10 (common logarithm) and 'e' (natural logarithm).
  • Common Logarithms: These use base 10 and are written as log(100) meaning log_{10}(100).
  • Natural Logarithms: These use the base 'e' (approximately 2.718), written as ln(20) meaning log_{e}(20).
Therefore, understanding the base helps in converting between exponential and logarithmic forms, simplifying the problem and making the solutions more intuitive.

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Most popular questions from this chapter

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\log _{a} m-\log _{a} n$$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 36$$

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=-2 x $$

$$ \left(\frac{4}{3}\right)^{x}=\frac{27}{64} $$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{0.006 x}=30 $$
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