Problem 25 - Precalculus : Building Concepts and Connections

Question

Evaluate each expression without using a calculator. $$\log 10^{k}$$

Step-by-Step Solution

Answer

The value of the expression \(\log 10^{k}\) is \(k\)

Step 1: Recognize logarithmic identity

The expression \(\log 10^{k}\) can be simplified using the base-10 logarithm rule. In general, for any positive number a ≠ 1 and any real number n, \(\log_a{a^n}=n\)

Step 2: Apply the identity to the expression

Applying the logarithmic identity to the expression \(\log 10^{k}\), the result will just be equal to \(k\).

Key Concepts

Logarithmic Identities

Logarithmic identities are mathematical properties that allow us to simplify expressions involving logarithms. Logarithms are the inverse operations of exponentials. One key identity is the power rule for logarithms:

• If you have a logarithmic expression of the form \(\log_a{a^n}\), it simplifies to just \(n\). This works because the base \(a\) raised to the power of \(n\) returns the original argument back inside the logarithmic function.

Another useful identity is the product rule:

• \(\log_a{(xy)} = \log_a{x} + \log_a{y}\)

This rule means that the logarithm of a product is the sum of the logarithms of its factors.
And then there's the quotient rule:

• \(\log_a{\left(\frac{x}{y}\right)} = \log_a{x} - \log_a{y}\)

This handy rule allows us to break down the logarithm of a quotient into the difference of the logarithms of the numerator and the denominator.
Understanding these properties is crucial for solving logarithmic problems efficiently.

Base-10 Logarithm

The base-10 logarithm, also known as the common logarithm, is commonly denoted as \(\log\) or \(\log_{10}\). It's particularly useful because it's based on the number 10, which is familiar from our everyday decimal system. In mathematical terms:

• \(\log_{10}(10^n) = n\)

This means that the logarithm of 10 raised to the power of \(n\) equals \(n\) itself. This property simplifies calculations involving powers of ten, making it a valuable identity in various scientific and engineering fields.Base-10 logarithms are often used in measuring the intensity of sound (decibels), the acidity of substances (pH), and earthquake magnitudes (Richter scale).
They are also used in information theory to calculate the information content in bits. By using base-10 logarithms, complex multiplicative relationships become more manageable linear ones.
When you apply logarithmic identities with base-10 logarithms, you can drastically reduce the complexity of solving logarithmic equations.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In simple terms, if you have a number, say \(10\), raised to an exponent \(k\), it means \(10\) multiplied by itself \(k\) times. This is written as:\[10^k\]The beauty of exponents lies in their simplicity and how they contribute to mathematical elegance. They provide a compact way of demonstrating calculations that would otherwise be extensive.Some important properties of exponents include:

• \(a^m \cdot a^n = a^{m+n}\): When you multiply like bases, add their exponents.
• \(\left(a^m\right)^n = a^{m\cdot n}\): When raising a power to another power, multiply the exponents.
• \(a^{-n} = \frac{1}{a^n}\): A negative exponent denotes the reciprocal of the base raised to the absolute value of the exponent.

Exponents are foundational in calculus, physics, and numerous other disciplines. They are directly linked to logarithms, as logarithms are simply the inverse operation of exponentiation.
By understanding how exponents work, you can better grasp how logarithms simplify complex multiplicative problems into manageable arithmetic ones.