Problem 26 - Algebra and Trigonometry

Question

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{5} x=4} & {\text { (b) } \log _{10} 0.1=x}\end{array} $$

Step-by-Step Solution

Answer

(a) \(x = 625\); (b) \(x = -1\).

Step 1: Understanding Logarithms and Equation (a)

The given equation is \( \log_5 x = 4 \). This indicates \( x \) is a number such that when 5 is raised to the power of 4, the result is \( x \). In mathematical terms, it can be expressed as \( x = 5^4 \).

Step 2: Solve (a) Using Exponential Form

Convert the logarithmic equation into an exponential form: \( x = 5^{4} \). Calculate \( 5^{4} \) which is \( 5 \times 5 \times 5 \times 5 = 625 \). Therefore, \( x = 625 \).

Step 3: Understanding Equation (b)

The given equation is \( \log_{10} 0.1 = x \). This means that 10 raised to the power of \( x \) equals 0.1. In mathematical terms, it can be expressed as \( 10^x = 0.1 \).

Step 4: Solve (b) Using Logarithmic Properties

Recognize that \( 0.1 = 10^{-1} \). Therefore, the equation \( 10^x = 10^{-1} \) can be solved by equating exponents, leading to \( x = -1 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. They are widely used in various fields such as science, engineering, and finance to model growth or decay processes. For example, if we have an exponential function expressed as \( y = a^x \), the base \( a\) is a constant and \( x \) is the variable exponent.
The properties of exponential functions include:

• The base \( a \) must be a positive real number other than 1.
• Exponential growth occurs when the base \( a \) is greater than 1.
• Exponential decay happens when \( a \) is between 0 and 1.

In our original exercise, when solving for \( \log_5 x = 4 \), we utilize these properties by converting the logarithmic equation to the exponential form: \( x = 5^4 \). This tells us \( x = 625 \), demonstrating how exponential functions work with constants and variables to provide solutions.

Logarithmic Equations

Logarithmic equations involve an equality that includes a logarithm of a variable. Understanding how to manipulate and solve these equations is crucial in calculus and algebra. A logarithmic equation generally has the form \( \log_b(y) = x \), where:

• \( b \) is the base of the logarithm.
• \( y \) is the argument or the quantity on which the logarithm operates.
• \( x \) is the exponent to which the base is raised to obtain \( y \).

To solve logarithmic equations, one technique is converting them into their exponential forms. For example, from the original problem \( \log_{10} 0.1 = x \), we translate this into the exponential equation: \( 10^x = 0.1 \). Recognizing that \( 0.1 \) equals \( 10^{-1} \), allows us to solve \( x = -1 \) by equating the exponents.

Base 10 and Base 5

Understanding different logarithmic bases is essential to solving problems involving logarithms. Base 10, known as the common logarithm, and base 5, seen in the problem, are examples of how different bases are applied.

Base 10 Logarithms:

• Commonly used for simplifying calculations in science and engineering.
• Expressed as \( \log_{10}(x) \), often simply written as \( \log(x) \).

Base 5 Logarithms:

• Less common but useful in specific contexts or exercises.
• Expressed as \( \log_{5}(x) \).

The approach to using different bases involves understanding their role in the corresponding exponential forms. For instance,

• The equation \( \log_5 x = 4 \) translates to \( x = 5^4 \).
• The equation \( \log_{10} 0.1 = x \) becomes \( 10^x = 0.1 \).

Through the base, we can grasp the concept of the problem and solve logarithmic equations efficiently.