Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{10} 0.0001\)

Step-by-Step Solution

Answer

-4

Step 1: Understand the Logarithm Notation

To evaluate the expression \(\log _{10} 0.0001\), bear in mind that \(\log_b a = c\) is equivalent to \(a = b^c\). This means we are looking for a number \(c\) satisfying the equation \(0.0001 = 10^c\).

Step 2: Simplify the Number to a Power of 10

0.0001 can be expressed as \(1 \times 10^{-4}\), which simplifies the equation to \(1 \times 10^{-4} = 10^c\).

Step 3: Solve for the Unknown Exponent

For the equation \(10^{-4} = 10^c\) to hold true, it can only be so if \(c = -4\), considering the fundamental belief in mathematics that if \(b^m = b^n\), then \(m = n\).

Key Concepts

Logarithmic Notation

Logarithmic notation is a different way to express exponentiation. It answers the question: To what power must the base be raised to produce a given number? For example, take the expression \(\log _{10} 0.0001\). The logarithm here asks us which exponent you would put on the base, which is 10 in this case, to get the number 0.0001.

Understanding this notation is crucial for evaluating logarithms. It can be translated into its equivalent exponential form: if \(\log_b a = c\), then \(a = b^c\). This means the log base \(b\) of \(a\) equals \(c\). Thus, when faced with a logarithmic expression, always remember that you're looking for the power that the base must be raised to in order to yield the particular number you're examining.

Exponentiation

Exponentiation is the process of raising a number (the base) to a certain power (the exponent), which denotes the number of times that the base is multiplied by itself. In the context of logarithmic expressions, exponentiation is the inverse operation to taking a logarithm.

For instance, if we have \(10^{-4}\), this indicates that 10 is multiplied by itself -4 times, which means we are actually dealing with the reciprocal of 10 raised to the 4th power, resulting in 0.0001. The ability to recognize and simplify expressions involving exponents is a key skill in solving logarithmic problems because it allows you to rewrite the equation in a simpler form, making the logarithm easier to evaluate.

Solving Logarithmic Equations

Solving logarithmic equations involves finding the value of the unknown exponent that makes the equation true. To do this, you'll often convert the logarithmic equation into its exponential form, allowing you to directly compare the exponents.

Following our example with the expression \(\log _{10} 0.0001\), we convert to an exponential equation: \(10^c = 0.0001\). Since we can rewrite 0.0001 as \(10^{-4}\), it becomes clear that \(c = -4\) because the bases are the same, and for the equation to hold true, the exponents must also be equal. This property simplifies the equation significantly and is the key to unlocking most logarithmic equations: when the bases are the same, their exponents must be as well to maintain equality.