Problem 57 - Precalculus : Building Concepts and Connections

Question

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$10^{\log (5 x)}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(10^{\log (5x)}\) is \(5x\).

Step 1: Identify the Expression

We need to evaluate or simplify: \(10^{\log (5 x)}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

The simplified form of the expression \(10^{\log (5x)}\) is \(5x\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial when simplifying logarithmic expressions. These properties are rules that allow us to manipulate logarithms in order to simplify the expression or solve logarithmic equations. One essential property is the inverse property, which states that the logarithm of a number to its own base equals the exponent. In other words, for any positive number x and base b, the expression \( b^{\log_b(x)} \) simplifies to \( x \).

This pivotal property can greatly simplify calculations without using calculators and is particularly helpful when dealing with expressions like \( 10^{\log(5x)} \).

Logarithm and Exponent Rules

Logarithms and exponents are intertwined concepts where the rules of one can often be applied to the other. This relationship is key in understanding how to manipulate and simplify expressions involving both. The logarithm of a number is the exponent to which the base must be raised to produce that number. Furthermore, the power rule for logarithms states that \( \log_b(x^y) = y \cdot \log_b(x) \), allowing us to move the exponent in front of the logarithm which can make the expression easier to simplify.

Another important exponent rule is that any number raised to the power of 1 is the number itself, such as in \( b^1 = b \). Utilizing these rules effectively means recognizing the base and its corresponding logarithm, making it possible to identify opportunities to apply them for simplification.

Logarithmic Simplification

Logarithmic simplification involves applying the properties of logarithms and rules of exponents strategically to reduce expressions to their simplest form. To simplify a logarithmic expression like \( 10^{\log(5x)} \) , you can observe that the base of the exponent and the base of the logarithm are the same (\(b = 10\)). By applying the inverse property of logarithms, which effectively 'cancels out' the logarithm and the exponent, the expression simplifies directly to \( 5x \).

When simplifying logarithmic expressions, it's also vital to remember that variables inside the log function must respect the domain of logarithmic functions. That is, the argument of the logarithm must be positive. Within the proper domain, simplification makes evaluating logs straightforward without the need for complex calculations or calculators.