Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, use the definition of common and natural logarithms to simplify. \(10^{\log (32)}\)

Step-by-Step Solution

Answer

The expression simplifies to 32.

Step 1: Recognize the Logarithmic Identity

Understand that the expression involves a common logarithm. The general identity is \( 10^{\log_{10}(x)} = x \). This means that when you have a base of 10 raised to the log of a number to the base 10, they simplify to the number itself.

Step 2: Simplify the Expression

Apply the identity from Step 1 to the given expression \( 10^{\log (32)} \). Since the logarithm is common (logarithm base 10), the expression simplifies directly to \( 32 \).

Key Concepts

Common Logarithms

Common logarithms are those logarithms that have a base of 10. They are often written simply as \( \log(x) \) rather than \( \log_{10}(x) \). This is because base 10 is commonly used and universally understood without needing explicit notation.
Common logarithms are handy in many fields, especially in sciences and engineering, because of their base 10, which makes them easier to compute mentally and more intuitive in a decimal number system.

• The typical identity for common logarithms is \( 10^{\log_{10}(x)} = x \). This means, any number x, when expressed in this form, simplifies to x itself.
• Common logarithms are used widely in solving exponential equations, especially where powers of ten are involved.

For the expression \( 10^{\log(32)} \), we used this identity to simplify it directly to 32, demonstrating the power and utility of common logarithms.

Natural Logarithms

Natural logarithms have a base \( e \), an irrational number approximately equal to 2.71828. They are denoted by \( \ln(x) \) instead of \( \log_e(x) \).

• Natural logarithms are incredibly significant in mathematics, especially in calculus, because they simplify expressions involving derivatives and integrals.
• This base \( e \) is selected for natural logarithms because it results in the simplest form when dealing with growth processes and complex mathematical equations.
• Just like common logarithms simplify exponential expressions of base 10, natural logarithms help simplify expressions of base \( e \).

Although natural logarithms weren't used directly in our exercise, understanding their behavior helps in recognizing their applications across various mathematical disciplines. They govern continuous growth and exponential decay, which is why they are prevalent in numerous natural processes and scientific calculations.

Logarithmic Identity

Logarithmic identities are fundamental rules that help simplify expressions involving logarithms. These core identities form the basis for solving more complex logarithmic equations.

• The identity \( 10^{\log_{10}(x)} = x \) used in the original exercise reflects the power of logarithms in reversing exponential situations. They are inverse functions, where the logarithm unwinds the exponential to retrieve the original number.
• Another crucial identity involves the natural logarithm: \( e^{\ln(x)} = x \), mirroring the same concept for base \( e \).
• Understanding and applying these identities is essential for simplifying logarithmic expressions or solving equations involving exponential and logarithmic functions.

These identities eliminate complexity and make seemingly complicated equations much more manageable, enabling us to handle problems efficiently, as illustrated in the exercise with the simplification of \( 10^{\log(32)} \) directly to 32.