Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$10^{\log 33}$$

Step-by-Step Solution

Answer

The value of the expression \(10^{\log 33}\) is 33.

Step 1: Identify the Base of the Logarithm and Exponent

First, note that the base of the logarithm and the base of the exponent are both 10. The expression is in the form \(10^{\log_{10} 33}\).

Step 2: Apply the Inverse Property of Logarithms

The inverse property of logarithms states that \(b^{\log_b x} = x\). This means we can simplify the expression \(10^{\log_{10} 33}\) to just 33.

Key Concepts

Evaluate Expressions

When asked to evaluate expressions, it's like being a detective where numbers and symbols are clues. In math, to evaluate is to find the value of an expression. It's vital to understand each component of the expression and how they relate to each other. For instance, expressions that include logarithms and exponents, like the example of evaluating \(10^{\log 33}\), require recognizing how to manipulate the represented numbers and operations effectively.

Understanding the inverse relationship between logarithms and exponents is crucial. When the base of the exponent and the base of the logarithm match, as they do in our example, the problem simplifies beautifully, like finding the missing piece of a puzzle. This process can be applied to more complex expressions as well, helping students become more skilled in handling a wide range of mathematical problems.

Logarithm Properties

Inverse Property of Logarithms

The inverse property of logarithms is an elegant example of mathematical symmetry. Logarithms are essentially exponents in disguise. When we have an expression like \( b^{\log_b x} \), what it literally means is that we're raising the base \( b \) to the power that would give us \( x \) when applied to the base \( b \) in a logarithmic function.

There's a direct switching role happening here: the logarithm is 'asking' what exponent we need for base \( b \) to get \( x \), and then the exponentiation uses that exact exponent to give us \( x \) back. It's like a game of mathematical ping-pong that always ends with the original value of \( x \). This property simplifies many problems and is a cornerstone when working with logarithms.

Exponentiation

Power Play of Numbers

Understanding exponentiation is critical in evaluating expressions involving powers. Exponentiation is when a number, known as the base, is multiplied by itself a certain number of times indicated by the exponent. For example, \(2^3\) means we multiply 2 by itself 3 times: \(2 \times 2 \times 2\), which equals 8.

With the expression we are evaluating, \(10^{\log 33}\), exponentiation is linked with the logarithm. The exponent is the logarithm of 33 with the base of 10. In essence, exponentiation is the 'undo' function of a logarithm. Grasping this gives students a powerful key to unlock a vast number of mathematical expressions and can lead to a deeper comprehension of functions and calculus as well.