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Chapter 11: Problem 95

Let \(f(x)=\log (x)\) a. Find \(f(0.01)\) b. Find \(a\) if \(f(a)=3\)

Short Answer

Expert verified
a) \( f(0.01) = -2 \), b) \( a = 1000 \)

Step by step solution

01

Understand the function

The function given is \( f(x) = \log(x) \). This is the logarithmic function with base 10.
02

Calculate \( f(0.01) \)

To find \( f(0.01) \), substitute \( x = 0.01 \) into the function: \[ f(0.01) = \log(0.01) \]. Since \( 0.01 = 10^{-2} \), we have \[ f(0.01) = \log(10^{-2}) = -2 \] because \( \log(10^{-2}) = -2 \).
03

Understand the problem for part b

We need to find \( a \) such that \( f(a) = 3 \). This means \[ \log(a) = 3 \].
04

Solve for \( a \)

To solve \( \log(a) = 3 \), rewrite the equation in its exponential form: \[ a = 10^3 \]. Therefore, \( a = 1000 \).

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Key Concepts

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

Logarithm Base 10
A logarithm answers the question: 鈥淭o what exponent must we raise a specified base to get a certain number?鈥� When we talk about logarithms, the base is very important.
The base-10 logarithm, also known as the common logarithm, uses 10 as its base. It is usually written as \(\text{log}(x)\).
For example, \(\text{log}(1000) = 3\) because \(\text{10}^3 = 1000\). Using the base-10 logarithm is very convenient in many calculations because it deals with powers of 10, which are easy to understand and work with.
So, if you see \(\text{log}(x)\), always remember it refers to the base-10 unless stated otherwise.
Evaluating Logarithms
Evaluating logarithms means finding the exponent that the base must be raised to obtain the given number. Here's a step-by-step breakdown:
  • Identify the base and the number you're dealing with. In most cases, for base-10 logarithms, the base is 10.
  • Rewrite the number as a power of the base. For example, 0.01 can be written as \(10^{-2}\).
  • Use the logarithmic function to find the exponent. So, \( \text{log}(0.01) = -2 \) because \(( 10^{-2} = 0.01 ).\)

As seen in the exercise, you substitute the given value into the function and apply the logarithmic rules to find the answer.
It's also vital to practice regularly and use various numbers to become comfortable with the process.
Exponential Form
Exponential form is a way of expressing numbers using a base and an exponent.
It's crucial when dealing with logarithms. To convert a logarithmic equation to an exponential form, use the relationship: \(\text{log}_b(a) = c\) translates to \(\text{b}^c = a \).
For instance, in the exercise, you needed to find \(a\) such that \( \text{log}(a) = 3. \)
  • Rewrite this as \(10^3 = a\) using the base-10 exponential form.
  • Calculate the exponent: \(10^3 = 1000\).

Thus, \(a = 1000\). The exponential form provides a straightforward method to find values and understand the relationship between logarithms and exponents.

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