Problem 61 - College Algebra and Calculus: An Applied Approach

Question

In Exercises \(61-90\), solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} x=4\)

Step-by-Step Solution

Answer

The solution to the equation \(\log _{10} x = 4\) is \(x = 10000\).

Step 1: Convert Logarithmic Form to Exponential Form

The initial given equation is \(\log _{10} x = 4\). To get rid of the logarithm, convert this equation into exponential form. Given that the base of the logarithm is 10, the result of the conversion would be \(10^4 = x\).

Step 2: Simplify the Exponential Expression

Now, evaluate \(10^4\). This results in 10000. So, the previous equation simplifies to \(x = 10000\).

Key Concepts

Exponential Form

To understand why we convert a logarithmic equation to exponential form, let's first clarify what these terms mean. In logarithmic form, \( \log_{b} a = c \), it means that \( b \) raised to the power of \( c \) gives \( a \):

• \( b \) is the base of the logarithm,
• \( a \) is the result of the base raised to the power \( c \),
• \( c \) is the exponent.

Thus, we can express the same relationship in exponential form as \( b^c = a \).
This conversion is crucial as it transforms the equation into a format that is more straightforward to solve, particularly when variables are inside the logarithmic expression. In our exercise, we converted \( \log_{10} x = 4 \) to its equivalent exponential form, \( 10^4 = x \).
Solving the transformed equation simplifies significantly, as in this example, where understanding that \( 10^4 \) equates to multiplying 10 by itself four times.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. This relationship is the basis for many of the properties that make logarithms useful in solving equations. When you see a logarithmic function such as \( \log_{10} x \), you are essentially asking "to what power must 10 be raised, to get \( x \)?"
This understanding helps demystify logarithmic equations. It is also why converting from logarithmic form to exponential form is a helpful step. Solving becomes more intuitive when you view it through the lens of exponents.

• Natural logs (\( \ln x \)) use the constant \( e \) as their base, while common logs, like our example, typically use base 10.
• Logarithms follow specific rules, such as the product, quotient, and power rules, which help simplify expressions and solve complex equations.

Building familiarity with these rules can aid in minimizing mistakes and effectively tackling various algebraic problems.

Algebraic Solutions

Algebraic solutions to logarithmic equations require mastering both the concepts of logarithms and exponents.
Once an equation is converted from logarithmic to exponential form, solving involves algebraic manipulations to isolate the unknown or variable.
In the example given, once we converted \( \log_{10} x = 4 \) to \( 10^4 = x \), we simplified the expression to find \( x = 10000 \).

• Inevitably, some problems might require additional algebraic steps, such as factoring quadratic equations or applying the properties of equality.
• Approximating the result may also be necessary, especially in more complex scenarios where exact solutions aren't easily obtainable.

Mastering algebraic techniques is crucial for solving these kinds of equations swiftly and accurately, providing a firm foundation for tackling various types of mathematical challenges.