Problem 15 - A Graphical Approach to Precalculus with Limits

Question

For each statement, write an equivalent statement in exponential form. Do not use a calculator. $$\log \sqrt{10}=0.5$$

Step-by-Step Solution

Answer

The equivalent exponential form is \( 10^{0.5} = \sqrt{10} \).

Step 1: Understand the Logarithmic Statement

The statement given is \( \log \sqrt{10} = 0.5 \). This means that the logarithm of \( \sqrt{10} \) to base 10 equals 0.5.

Step 2: Relate Logarithmic Form to Exponential Form

We know the general logarithmic form is \( \log_b a = c \), which means \( b^c = a \). Here, \( b \) is the base of the logarithm, which is 10, \( a \) is the number inside the log, \( \sqrt{10} \), and \( c \) is the result, 0.5.

Step 3: Apply the Exponential Relationship

Using the formula from Step 2, express the statement \( \log \sqrt{10} = 0.5 \) in exponential form: \( 10^{0.5} = \sqrt{10} \).

Step 4: Verify the Exponential Form

Verify that \( 10^{0.5} \) indeed equals \( \sqrt{10} \). Since \( 0.5 \) is equivalent to a square root in exponential terms, \( 10^{0.5} = \sqrt{10} \), confirming the statement.

Key Concepts

Exponential Form

In mathematics, understanding the connection between exponential and logarithmic forms is essential for solving equations and expressing relationships. The exponential form is a way to represent equations where a number is raised to a power.

For example, when we have a base number and an exponent, like \( b^c = a \), it illustrates how many times the base \( b \) is multiplied by itself to reach the value \( a \). This is known as the exponential form.

• "Base" is the number being multiplied.
• "Exponent" indicates how many times multiplication occurs.
• The result is the "power" or "product."

Converting a logarithmic equation such as \( \log \sqrt{10} = 0.5 \) to exponential form involves understanding that the exponent (0.5) indicates the power needed to turn 10 into \( \sqrt{10} \). The relationship \( 10^{0.5} = \sqrt{10} \) is a simple demonstration of this principle.

Logarithmic Form

The logarithmic form is another way to express relationships between numbers, especially those involving exponential calculations. It is the inverse operation of an exponentiation and answers the question, "To what power must a base be raised to obtain a certain number?"

In the general form \( \log_b a = c \):

• \( b \) is the base of the logarithm.
• \( a \) is the number you want to find the logarithm for.
• \( c \) is the power the base must be raised to yield \( a \).

Taking the equation \( \log \sqrt{10} = 0.5 \) from the original exercise, \( 10 \) is the base, \( \sqrt{10} \) is the value we are taking the logarithm of, and \( 0.5 \) is the power. Converting from logarithmic to exponential form directly shows us the relationship: the base 10 raised to 0.5 equals \( \sqrt{10} \). This highlights the elegance and utility of logarithms in breaking down complex exponential expressions.

Base-10 Logarithm

Base-10 logarithms, also known as common logarithms, are widely used in several fields, including science and engineering. They are particularly useful because they rely on the number 10, which is the foundation of our decimal system.

When a logarithm is written without specifying a base, it is typically assumed to be base 10. So, \( \log \sqrt{10} \) is shorthand for \( \log_{10} \sqrt{10} \). This helps simplify expressions and calculations by taking advantage of the properties of powers of 10.

• Base-10 logarithms simplify calculations, especially in scientific notation.
• Base-10 is intrinsic to our standard number system, making these logarithms intuitive.

In the context of the exercise, understanding the base-10 logarithm makes translating \( \log \sqrt{10} = 0.5 \) into \( 10^{0.5} = \sqrt{10} \) straightforward. This exemplifies how logarithms serve as a bridge between an exponential value and its base powers, especially when working with base-10.