Problem 27 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \log 0.0001 $$

Step-by-Step Solution

Answer

The exact value is -4.

Step 1: Convert to Exponential Form

To find the exact value of a logarithm, we often start by rewriting the logarithmic equation in its equivalent exponential form. The problem asks for \( \log 0.0001 \). We need to find the power to which 10 must be raised to get 0.0001. In other words, we rewrite it as \( 10^x = 0.0001 \).

Step 2: Express Decimal as a Power of 10

Next, express the decimal number 0.0001 as a power of 10. The number 0.0001 is equivalent to \( 10^{-4} \) because moving the decimal four places to the right converts it to 1, and each move corresponds to a negative power of ten.

Step 3: Set the Logarithmic Equation

From the exponential expression \( 10^x = 10^{-4} \), we see that the exponents must be equal since the bases are the same. Thus, \( x = -4 \).

Step 4: Interpret the Logarithm

The equation \( \log 0.0001 = x \) becomes \( \log 0.0001 = -4 \). Therefore, the exact value of the logarithm \( \log 0.0001 \) is \(-4\).

Key Concepts

Exponential Form

Understanding the concept of exponential form is crucial when dealing with logarithms. When a logarithm is written, such as \( \log_{10}(y) = x \), it is expressing the idea that 10 raised to the power of \( x \) results in \( y \). In the context of the exercise, finding \( \log 0.0001 \) requires restating the problem as an exponential equation: \( 10^x = 0.0001 \). This equation asks us to determine what power \( x \) should be applied to 10 to produce 0.0001.

Learning to convert between logarithmic and exponential forms enables students to handle logarithmic problems more effectively. The conversion shows the relationship between multiplication and repeated addition, which plays a fundamental role in understanding many mathematical functions.

Here are some simple steps to switch from logarithmic form to exponential form:

• Identify the base of the logarithm, which is the number being raised to a power.
• Determine the result or argument of the logarithm, which is the number you are trying to reach using the base.
• Write the equation: base raised to the power of the logarithm equals the argument.

Decimal to Power of 10

Decimals can be expressed as powers of 10, which is a handy trick in logarithms. When working with small decimal numbers, like 0.0001, it can be helpful to see them in this form. The number 0.0001 can be rewritten as \( 10^{-4} \).

How do we get to \( 10^{-4} \)? Each time you move the decimal point one place to the right, you multiply the number by 10. So, moving the decimal four places to make 0.0001 into 1 requires multiplying it by 10 four times, or in other words, multiplying by \( 10^4 \). But since 0.0001 is much smaller than 1, we actually need to divide by \( 10^4 \), which is the same as multiplying by \( 10^{-4} \).

Here is a quick way to convert decimals to powers of 10:

• Count how many places the decimal needs to move to turn the number into 1.
• If moved to the left, count as a negative exponent; if moved to the right, count as a positive exponent.
• Express the number as \( 10^{\text{-(number of places)}}\).

Negative Exponents

Negative exponents are used to denote reciprocals or fractions in terms of powers of 10. A number like \( 10^{-n} \), where \( n \) is a positive integer, indicates that 1 is divided by \( 10^n \). This is a powerful concept in mathematics, helping us express very small numbers compactly and manipulate them with ease.

For our given problem, when we state that \( 0.0001 = 10^{-4} \), we are saying it is the result of dividing 1 by \( 10^{4} \), or \( \frac{1}{10000} \). Working with negative exponents allows us to easily convert between different numerical forms for effective problem-solving.

Let's affirm why this concept is important:

• Negative exponents simplify mathematical operations with very small numbers.
• They help in accurately denoting measurement units in sciences like chemistry and physics.
• They make it easier to derive the logarithm of small numbers by equating the expression to an exponential form.
• Understanding them enhances the ability to interpret and solve algebraic expressions.