Problem 37 - College Algebra

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: Understanding the Expression

We need to simplify the expression \(10^{\log (32)}\). A key point to consider here is the relationship between logarithms and exponents.

Step 2: Using Logarithm Principles

Recall that for a function \(f(x) = 10^{\log_{10}(x)}\), the function is essentially \(x\). This means \(10^{\log_{10}(32)} = 32\).

Step 3: Applying the Property

Apply the property: For any positive number \(a\), the expression \(a^{\log_a(b)} = b\). Here, \(a = 10\), so \(10^{\log_{10}(32)} = 32\).

Step 4: Simplifying the Expression

With the property applied, we have simplified \(10^{\log(32)}\) to simply 32.

Key Concepts

Common Logarithm

A common logarithm, often denoted as \( \log(x) \), is a logarithm with base 10. This is the logarithm you frequently find on a scientific calculator, usually shown simply as "log." The definition of a common logarithm is both simple and elegant: it tells us the power (or exponent) to which we must raise the number 10 to obtain the value \( x \).

• For instance, \( \log(100) = 2 \) because \( 10^2 = 100 \).
• Similarly, \( \log(10) = 1 \) because \( 10^1 = 10 \).

Understanding common logarithms helps when dealing with equations in the form of exponentials where the base is 10. It simplifies calculations and transforms multiplication into addition, which is less computationally intensive. This usability makes them indispensable in science and engineering.

Natural Logarithm

The natural logarithm is a logarithm that has the constant \( e \) (approximately 2.718) as its base. It is denoted as \( \ln(x) \). Natural logarithms are particularly useful in continuous growth and decay processes, often found in calculus involving exponential functions.
Natural logarithms transform multiplicative processes into additive ones, similar to other logarithms but are particularly efficient with the base \( e \) due to its natural occurrence in various mathematical contexts.

• Remember: \( \ln(e) = 1 \) because the power to which \( e \) must be raised to get \( e \) is 1.
• Also, \( \ln(1) = 0 \) since any number raised to the power of 0 is 1.

Their properties make them very important in higher mathematics, especially when dealing with derivatives and integrals of exponential functions.

Exponential Functions

Exponential functions are a class of mathematical functions where the variable is in the exponent. These functions can be expressed generally as \( f(x) = a^x \), where \( a \) is a constant called the base and \( x \) is the exponent.
Exponential functions are vital in depicting rapid growth or decay processes, like compounding interest or population growth.

• A classic example is \( 2^x \), demonstrating exponential growth as \( x \) increases.
• Conversely, \( 2^{-x} \) shows exponential decay.

A special feature of exponential functions is their inverse relationship with logarithmic functions. This relationship is crucial as it allows one to solve exponential equations using logarithms and vice versa, shown in calculations often reduced by using the property \( a^{\log_a(b)} = b \).Understanding how exponential functions transform and operate is fundamental in fields like physics, finance, and other sciences that model real-world phenomena using mathematics.