Problem 20 - Algebra for College Students

Question

Write each logarithmic statement in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{10} 0.000001=-6 $$

Step-by-Step Solution

Answer

The exponential form is \(10^{-6} = 0.000001\).

Step 1: Identify the Base of the Logarithm

In the logarithmic statement given, \( \log_{10} 0.000001 = -6 \), the number 10 is the base of the logarithm. This is because we are dealing with a common logarithm, which has a base of 10.

Step 2: Recognize the Relationship

The equation \( \log_{10} 0.000001 = -6 \) implies that the logarithm represents an exponent. Specifically, it tells us that 10 raised to some power equals 0.000001.

Step 3: Convert to Exponential Form

Rewrite the logarithmic statement into its equivalent exponential form. The equation \( \log_{10} 0.000001 = -6 \) transforms to \( 10^{-6} = 0.000001 \). This means that when 10 is raised to the power of -6, it results in 0.000001.

Key Concepts

Understanding Common Logarithms

Logarithms are important in mathematics as they help simplify complex multiplication problems. A common logarithm is a logarithm with a base of 10. When we see the notation \( \log_{10} \), we are dealing with a common logarithm implicitly, often written simply as \( \log \). The base 10 is widely used because it aligns with the decimal system, making it intuitive and easy to use.

• The logarithm \( \log_{10} \) tells us what power we need to raise 10 to get a specific number.
• Common logarithms are often used in scientific fields and settings where we measure phenomena on a multiplicative scale like sound intensity and pH levels.
• When no base is noted in a logarithm, it is usually understood to be base 10, reinforcing its "common" status.

Understanding common logarithms allows us to effectively translate between logarithmic and exponential forms, enabling easier calculations and problem-solving.

Exploring Exponential Equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. These equations are prevalent in growth and decay problems, such as population growth or radioactive decay, where quantities increase or decrease exponentially.

• Exponential equations can sometimes seem complex, but simplifying them often makes understanding much easier.
• Simplifying involves breaking the equation into recognizable patterns, like converting logarithmic statements into an exponential form to reveal the underlying relationship.
• For example, converting \( \log_{10} 0.000001 = -6 \) into \( 10^{-6} = 0.000001 \) shows how exponential equations express relationships.

These conversions provide insights into how exponential equations work, demonstrating how different values relate through powers and bases.

Deciphering the Base of a Logarithm

The base of a logarithm indicates the number that is repeatedly multiplied. In the logarithmic expression \( \log_{b} a = c \), \( b \) is the base, \( a \) is the number you are taking the logarithm of, and \( c \) is the exponent.

• Understanding the base is essential for converting a logarithmic form to its exponential equivalent.
• The base tells you what number you are repeatedly multiplying to reach \( a \).
• In common logarithms, the base is always 10, making conversions straightforward for most practical purposes.

Recognizing the base of a logarithm is key to solving logarithmic and exponential problems, as it helps outline the structure needed to switch between forms efficiently.