Problem 22 - Intermediate Algebra

Question

Fill in the blanks. a. We read \(\log _{5} 25\) as "log,____ 5 ____ 25.” b. We read \(\log x\) as " ______ of \(x\) ".

Step-by-Step Solution

Answer

a. "log, base 5 of 25." b. "logarithm of \( x \)."

Step 1: Understanding Logarithmic Notation

In mathematics, the notation \( \log_b a \) is used to find the power to which the base \( b \) must be raised to get the number \( a \). This is known as the logarithm of \( a \) to the base \( b \). For instance, in \( \log_5 25 \), 5 is the base and 25 is the power to which 5 must be raised to get 25.

Step 2: Reading 'logarithm' Expressions

The expression \( \log_b a \) is typically read aloud as "log base \( b \) of \( a \)." So, the correct way to read \( \log_5 25 \) is "log base 5 of 25." This is because the term "base" refers to the number that is being raised to a power.

Step 3: Understanding Common Logarithms

The notation \( \log x \) is a shorthand for the common logarithm, usually implying base 10 or the natural logarithm with a given base if context is provided. It's often read aloud as "logarithm of \( x \)." In the absence of a specified base, it is generally understood to be 10.

Step 4: Fill in the Blanks

a. We read \( \log _{5} 25 \) as "log, base 5 of 25." b. We read \( \log x \) as "logarithm of \( x \)."

Key Concepts

Logarithmic Notation

Logarithmic notation is a way to express the relationship between numbers in the form of a logarithm. The basic idea is to figure out how many times we must multiply a base number to get another number. For example, in the expression \( \log_b a \), we are asking: "What power must we raise \( b \) to in order to get \( a \)?" Here, \( b \) is called the "base," and \( a \) is the result of raising the base to some power. Understanding this notation helps us analyze and work with exponential growth or decay effectively. Key points about logarithmic notation:

• The base \( b \) must always be a positive number and not equal to 1.
• If \( \log_b a = c \), then \( b^c = a \). This is the fundamental definition of a logarithm.
• Logarithms are the inverse operation of exponentiation.

Breaking down these expressions helps us solve problems in a systematic way.

Common Logarithms

Common logarithms are a specific type of logarithm with a base of 10. When mathematicians refer to \( \log x \) without explicitly stating a base, they generally mean \( \log_{10} x \). This is because the base 10 is commonly used in computations and is understood by default.Common logarithms are especially useful in:

• Scientific calculations, where measurements often span several orders of magnitude.
• Engineering contexts, where base-10 measurements assist in standardizing calculations.
• The Richter scale in seismology, where it helps to measure the magnitude of earthquakes.

The common logarithm simplifies expressions by reducing large numbers into manageable ones. For instance, converting the large number 1,000 to a log scale just yields 3 because \( 10^3 = 1,000 \). By default, if you don't see a base written, you can assume it's 10.

Reading Logarithmic Expressions

Reading logarithmic expressions involves understanding how to verbalize the mathematical notation so that it is clear and precise. For the expression \( \log_b a \), you would say "log base \( b \) of \( a \)." This form of expression helps in communicating mathematical ideas clearly.Let's break it down:

• The word "log" simply indicates that we're talking about a logarithmic function.
• The "base \( b \)" part tells us which number is being repeatedly multiplied to reach the number \( a \).
• "Of \( a \)" indicates the specific number that results from raising the base to a certain power.

By consistently using these terms, students can more easily discuss and solve logarithmic problems. A good exercise is to practice reading and writing various logarithmic expressions to build familiarity and confidence.