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Chapter 10: Problem 9

Write in logarithmic form. $$10^{-3}=0.001$$

Short Answer

Expert verified
\( \text{log}_{10}(0.001) = -3 \)

Step by step solution

01

Identify the Parts of the Given Exponential Equation

Recognize the base, the exponent, and the result in the exponential equation. In this case, the base is 10, the exponent is -3, and the result is 0.001.
02

Recall the Logarithmic Form

The logarithmic form of an exponential equation is given as: \( \text{log}_b(y) = x \), where \( b \) is the base, \( x \) is the exponent, and \( y \) is the result.
03

Rewrite the Exponential Equation in Logarithmic Form

Using the exponential equation \( 10^{-3} = 0.001 \) and the format \( \text{log}_b(y) = x \), rewrite it as \( \text{log}_{10}(0.001) = -3 \).

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Key Concepts

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

logarithms
Understanding the basics of logarithms is crucial to solving exponential equations.
A logarithm answers the question: To what exponent must we raise a given base to obtain a certain number? For example, in the logarithmic equation \(\text{log}_{10}(0.001)=-3\), we are asking: To what power must 10 be raised to get 0.001?
  • The base here is 10.
  • The result is 0.001.
  • The exponent is -3.
Logarithms are incredibly useful in various fields such as science, engineering, and finance. They help to transform multiplicative relationships into additive ones, simplifying complex calculations. Remember, when you see a log operation, you鈥檙e working out an exponent or power.
exponential equations
Exponential equations, like \(10^{-3}=0.001\), have variables in the exponent.
They are commonly written in the form \(b^x=y\), where \(b\) is the base, \(x\) is the exponent, and \(y\) is the result.
For conversion to logarithmic form, we need to understand that each component of this equation plays a specific role:
  • The base (\(b\)) is the number that gets raised to the exponent.
  • The exponent (\(x\)) is the power to which the base is raised.
  • The result (\(y\)) is the outcome of the base raised to the power.
To convert the exponential form \(b^x=y\) to logarithmic form, rewrite it as \(\text{log}_b(y)=x\). This process helps in solving equations where the exponent is the unknown.
base of logarithm
The base in a logarithm is a fundamental component that defines the logarithmic operation.
In the equation \(\text{log}_{10}(0.001) = -3\), the base is 10.
Bases can vary, such as 2, 10, or even Euler's number \(e\). Here's what each element means:
  • The base (10) is the number we repeatedly multiply.
  • The result (0.001) is what we get after applying the base to the exponent.
  • The exponent (-3) is how many times we use the base in the multiplication process, including any negative or fractional exponents.
Understanding the base helps to decode the logarithmic operations accurately. It indicates the number that you are repeatedly multiplying to express the result.

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Most popular questions from this chapter

Solve each equation. Give exact solutions. $$ \log _{4}(2 x+8)=2 $$

How much money must be deposited today to amount to \(\$ 1000\) in \(10 \mathrm{yr}\) at \(5 \%\) compounded continuously?

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{-x}=\pi $$

Find the amount of money in an account after 8 yr if \(\$ 4500\) is deposited at \(6 \%\) annual interest compounded as follows. (a) Annually (b) Semiannually (c) Quarterly (d) Daily (Use \(n=365 .)\) (e) Continuously

Solve each equation. Approximate solutions to three decimal places. $$ 9^{-x+2}=13 $$
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