Problem 91 - College Algebra

Question

Describe the following property using words: \(\log _{h} b^{x}=x.\)

Step-by-Step Solution

Answer

The expresson \(\log _{h} b^{x}=x\) can be described as: The logarithm of \(b^{x}\) to the base \(h\) equals \(x\). This means when you raise \(h\) to the power \(x\), you get \(b^{x}\).

Step 1: Recognize the logarithm expression

Firstly, recognize the given expression \(\log _{h} b^{x}=x\). Here \(\log _{h} b^{x}\) denotes the logarithm of \(b^{x}\) to the base \(h\). The output of this log operation is \(x\).

Step 2: Apply logarithm properties

Logarithm property states that for any two positive real numbers \(b\) and \(h\) (where \(h ≠ 1\)), the logarithm of \(b^{x}\) to the base \(h\) is \(x\), i.e., \(\log_{h}(b^{x}) = x\). This means, \(b^{x}\) equals \(h\) raised to the power \(x\). In other words, \(h\) multiplied by itself \(x\) times equals to \(b^{x}\).

Step 3: Interpret the expression in words

Lastly, interpret the expression \(\log _{h} b^{x}=x\) in words. The given equation is equivalent to saying 'The logarithm of \(b^{x}\) to the base \(h\) is \(x\)', which implies that, if you raise \(h\) to the power of \(x\), you will get \(b^{x}\).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number, called the base, to the power of an exponent. It is a shorthand way to represent repeated multiplication. For example, when we see the expression \( b^x \), \( b \) is the base and \( x \) is the exponent. This means we multiply \( b \) by itself a total of \( x \) times.

• If \( x = 2 \), then \( b^x \) is \( b \) multiplied by itself once (i.e., \( b \times b \)).
• If \( x = 3 \), \( b^x = b \times b \times b \).

Exponentiation is the inverse operation of taking a logarithm. If you know the result of \( b^x \) and the base \( b \), you can find \( x \) by using logarithms. Understanding this concept is crucial for solving problems involving exponential growth or decay, among other applications.

Base of Logarithm

The base of a logarithm is essential for understanding logarithms. In the expression \( \log_h b \), \( h \) is the base. It indicates what number you are repeatedly multiplying by itself to reach a certain value. When we talk about \( \log_h b \), we essentially mean: 'To what power must \( h \) be raised, to produce \( b \)?'

• The base \( h \) should always be a positive real number, different from 1.
• The choice of base can affect interpretations of logarithmic calculations significantly.
• Common bases include 10 (common logarithm), \( e \) (natural logarithm), and 2 (binary logarithm).

Choosing a specific base can simplify calculations in different mathematical contexts. This makes understanding the base crucial when interpreting logarithmic expressions.

Logarithmic Expression

A logarithmic expression, such as \( \log_h b \), is a powerful mathematical tool used to denote the power that a base must be raised to, in order to obtain a specific number. Logarithms transform multiplicative relationships into additive ones, making them particularly useful in scientific and engineering contexts where calculations involve exponential growth or decay.When you read \( \log_h b \), you should understand it as saying 'What power does \( h \) require to become \( b \)?' In practical terms:

• \( \log_h b = x \) implies \( h^x = b \).
• This means if \( h \) raised to the power \( x \) equals \( b \), then \( x \) is your solution to the logarithmic expression.

Logarithms link directly back to exponents, providing an elegant balance between these two mathematical operations. Mastering their concepts can greatly simplify the complexity of many problems.