Problem 86 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log 4 x=2$$

Step-by-Step Solution

Answer

The solution is \(x = 25\).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log 4x = 2 \). This implies that the base 10 logarithm of \(4x\) is equal to 2. In other words, 10 raised to the power of 2 equals \(4x\).

Step 2: Convert Logarithmic to Exponential Form

To find \(x\), we need to convert the logarithmic equation to its exponential form. The equation \(\log 4x = 2\) means \(10^2 = 4x\).

Step 3: Solve for \(x\)

Now, calculate \(10^2\), which is 100. So, we have the equation \(100 = 4x\). To find \(x\), we divide both sides by 4: \(x = \frac{100}{4}\).

Step 4: Calculate \(x\)

Divide 100 by 4 to get \(x = 25\). Thus, the solution for \(x\) in the equation \(\log 4x = 2\) is 25.

Key Concepts

Exponential Form

Converting a logarithmic equation to exponential form is a crucial step in solving logarithmic equations. The original equation is given in logarithmic form: \( \log 4x = 2 \). To understand what this means, remember that logarithms answer the question: 'To what power must the base be raised to produce a given number?' Here, the base is 10, a common logarithm, and the given number is \(4x\).
The equation states that the base 10 raised to the power of 2 will result in 4 times \(x\). This transformation is expressed as \(10^2 = 4x\), which is the exponential form.

• Logarithmic form \( \log_b A = C \)
• Exponential form \( b^C = A \)

Switching from an equation in logarithmic form to exponential form makes it easier to solve for \(x\), since exponential equations are often more straightforward to manipulate algebraically.

Solving Equations

Once the logarithmic equation \( \log 4x = 2 \) is converted into its exponential form \(10^2 = 4x\), solving for \(x\) becomes much simpler. Now you solve an algebraic equation rather than dealing with logarithms. The calculation involves straightforward arithmetic:
1. Calculate \(10^2\), which equals 100.
2. Substitute this back into the equation: \(100 = 4x\).
3. To isolate \(x\), divide both sides by 4:
\[ x = \frac{100}{4}\]4. Simplify the resulting expression to find \(x = 25\).
This step-by-step breakdown shows that even complex-looking logarithmic equations can be easily managed with simple operations once in exponential form. Breaking down each arithmetic operation ensures accuracy and clarity in finding the solution.

Base 10 Logarithm

The concept of a base 10 logarithm, also known as the common logarithm, is fundamental in many equations. It is often simply noted as \( \log \) without specifying the base, assuming it's 10. In the equation \( \log 4x = 2 \), the base is 10.
Understanding base 10 logarithms can simplify math involving large numbers, as they help assess power levels more easily. For instance, knowing \( \log 100 = 2 \) intuitively tells us it's because 100 is 10 squared.

• Base 10: Allows us to relate logarithms directly to powers of ten.
• Used extensively in science and engineering to manage very large or very small numbers.

In solving \( \log 4x = 2 \), recognizing the role of base 10 helps transition the equation into exponential form readily, making further calculus operations more approachable.