Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} 2 x=7\)

Step-by-Step Solution

Answer

The solution is \(x = 5,000,000\)

Step 1: Convert Equation from Logarithmic Form to Exponential Form

The given equation is \(\log _{10} 2x = 7\). In exponential form, this becomes \(10^7 = 2x\)

Step 2: Solve the Equation for x

To isolate \(x\), divide both sides of the equation by 2. The equation becomes \(x = (10^7) / 2\)

Step 3: Calculate the Value of x and Round to 3 Decimal Places

After division, evaluate the expression to get \(x\). After rounding to three decimal places, \(x = 5,000,000\)

Key Concepts

Logarithmic Form

Logarithmic form is used to express the power to which a base number is raised to obtain a certain value. A general representation is

• \(\log_b(a) = c\)

where \(b\) is the base of the logarithm, \(a\) is the result of the base raised to the power of \(c\), and \(c\) is the exponent or logarithm. In our exercise, the equation \(\log_{10}(2x) = 7\) suggests that 10 must be raised to the power 7 to get 2x. This form is extremely useful in simplifying the equation before solving it by converting it into an exponential form.

Exponential Form

Exponential form is the counterpart to logarithmic form and is often more straightforward to solve. This involves rewriting a logarithmic equation so that the base is raised to a power, and it equals a known result.

• The conversion goes from \(\log_b(a)=c\) to \(b^c = a\).

For the given example \(\log_{10}(2x) = 7\), the exponential form is \(10^7 = 2x\). This equation is simpler to manipulate because you can directly calculate \(10^7\) to find the value of \(2x\) and solve for \(x\). Knowing how to switch between forms allows you to work with different types of problems efficiently.

Rounding Numbers

Rounding numbers is a method used to estimate values to make them simpler and easier to interpret, especially when a high level of precision is unnecessary. It's crucial in reporting numerical data succinctly, particularly in contexts like mathematics and science, where specific numerical precision is not always essential. In this exercise, we approximate the solution to three decimal places.

• First, after calculating \(10^7 = 10,000,000\), we divide by 2 to find \(x = 5,000,000\).
• Because no decimal places are present in \(5,000,000\), rounding to three decimal places leaves the result unchanged.

Even though the rounding in this instance seems trivial, it illustrates how rounding works and reinforces the process used to approach rounding in different contexts effectively.