Problem 15 - Intermediate Algebra

Question

For Problems \(11-20\), write each of the following in exponential form. For example, \(\log _{2} 8=3\) becomes \(2^{3}=8\) in exponential form. $$ \log _{10} 10,000=4 $$

Step-by-Step Solution

Answer

The exponential form is \(10^4 = 10,000\).

Step 1: Understanding the Logarithmic Expression

The given expression is \( \log_{10} 10,000 = 4 \). This means that 10 must be raised to a power to get 10,000, and that power is 4.

Step 2: Converting to Exponential Form

To convert the logarithmic expression \( \log_{10} 10,000 = 4 \) into exponential form, rearrange it based on the definition of logarithms: \( b^y = x \), where \( \log_b x = y \). Here, \( b = 10 \), \( y = 4 \), and \( x = 10,000 \).

Step 3: Write the Exponential Equation

According to the conversion, we rewrite the expression in exponential form as \( 10^4 = 10,000 \). This shows how 10 raised to the 4th power equals 10,000.

Key Concepts

Logarithmic Expression

Understanding a logarithmic expression can be quite straightforward once you get the hang of it. A logarithmic expression generally follows the pattern \( \log_b x = y \). This means you're trying to find the power \( y \), called the logarithm, to which the base \( b \) must be raised to produce \( x \). In simple terms, a logarithm answers the question, "To what power must the base be raised to obtain the given number?"

This is why logarithms are considered the inverse of exponentiation. For example, if we take the expression \( \log_{10} 10,000 = 4 \), it tells us that 10 raised to the power of 4 equals 10,000.

• Base: The number being repeatedly multiplied (10 in our example).
• Argument: The number you want to achieve by raising the base (10,000).
• Logarithm: The power/exponent (4 in this case).

Let's move on to understanding how we can rearrange this into a more intuitive format using exponential form.

Conversion to Exponential Form

Converting from a logarithmic form to an exponential form simplifies the expression by translating it into something many find easier to understand. The general format \( \log_b x = y \) can be rewritten in exponential form as \( b^y = x \). This rearrangement directly shows the relationship between the base, exponent, and result.

For our specific example, \( \log_{10} 10,000 = 4 \), converting to exponential form involves:

• Identifying the base \( b \) as 10.
• Identifying the exponent \( y \) as 4.
• Identifying the result \( x \) as 10,000.

When rearranged, you write this as \( 10^4 = 10,000 \). This means we've converted the logarithmic expression into a clearer statement that shows us directly how the numbers relate through repeated multiplication.

Powers of Ten

Powers of ten are a fantastic way to quickly comprehend large numbers, as well as the process of exponentiation in general, since they utilize the base of 10. Each exponent on base 10 represents how many times to multiply 10 by itself.

Let's break down the example \( 10^4 = 10,000 \).

• 10¹: Just 10.
• 10²: Ten multiplied by itself, resulting in 100.
• 10³: A further multiplication, equaling 1,000.
• 10⁴: Finally, multiply once more to get 10,000.

Using powers of ten makes it easier to understand and visualize the principle of multiplying large numbers. This is why this process is heavily used in scientific notation, helping to simplify calculations and to write large numbers more concisely.