Problem 46 - Intermediate Algebra

Question

Solve each equation. $$ \log (2-x)=3 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -998 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log(2-x) = 3 \). This means that \( (2-x) \) is the argument of the logarithmic function, and we need to find \( x \) for which this equation holds true.

Step 2: Convert Logarithmic to Exponential Form

Recall that if \( \log_b(a) = c \), this is equivalent to saying \( b^c = a \). In this problem, we have a common (base 10) logarithm, so we rewrite the equation as \( 10^3 = 2-x \).

Step 3: Simplify Exponential Equation

Calculate \( 10^3 \), which is 1000. Substitute this in the equation from Step 2, resulting in \( 1000 = 2-x \).

Step 4: Solve for x

Rearrange \( 1000 = 2-x \) to solve for \( x \). Subtract 2 from both sides to get \( 998 = -x \). Finally, multiply through by -1 to solve for \( x \), giving \( x = -998 \).

Key Concepts

Exponential Form

Understanding the exponential form is essential for solving logarithmic equations. Let's start with what a logarithm represents. When you see an equation like \( \log_b(a) = c \), it translates into an exponent in the form of \( b^c = a \). Here, \( b \) is the base of the logarithm, \( c \) is the exponent, and \( a \) is the result of raising the base to the power of the exponent.

In our specific problem, we deal with a common logarithm where the base is 10, as indicated by \( \log(2-x) = 3 \). Converting this into exponential form means expressing it as \( 10^3 = 2-x \). This step is crucial because it transforms a potentially complex logarithmic problem into a more familiar algebraic one.

• Logarithmic form: \( \log_{10}(2-x) = 3 \)
• Exponential form: \( 10^3 = 2-x \)

Understanding these conversions allows us to work more comfortably with logarithms, treating the equation more like a straightforward algebra problem.

Common Logarithms

Common logarithms, or base 10 logarithms, are one of the most frequently encountered types of logarithms, often used in scientific calculations and everyday mathematical problems. They are conveniently written as \( \log \) without a base to signify that they use base 10 automatically.

With a common logarithm, you're calculating the exponent needed to raise 10 to attain a particular number. For example, to find the common log of 1000, you're asking: "10 raised to what power equals 1000?" The answer is 3, because \( 10^3 = 1000 \).

• Common logarithms are base 10 by default.
• In calculations: \( \log(1000) = 3 \), meaning \( 10^3 = 1000 \).

Common logarithms simplify our task by making calculations with their universal base, 10, easier, and they appear naturally in fields from physics to finance.

Solving Equations Step-by-Step

Breaking down equations and solving them step-by-step helps ensure we don't miss any small details. With logarithmic equations, this methodical approach can be especially handy.

Consider our example: \( \log(2-x) = 3 \).

• First, convert the logarithmic to an exponential form: \( 10^3 = 2-x \).
• Calculate \( 10^3 \) resulting in 1000.
• Set this equal to \( 2 - x \), as in \( 1000 = 2 - x \).
• Rearrange the equation to solve for \( x \). Subtract 2 from both sides to get \( 998 = -x \).
• Multiply through by -1. Thus, \( x = -998 \).

Following these steps, we arrive at our solution. Solving step-by-step ensures clarity and precision, allowing us to tackle complex equations systematically.