Problem 96 - Elementary and Intermediate Algebra

Question

a. \(\log x-\log (x+7)=-1\) b. \(\log x-\log (x+7)=1\)

Step-by-Step Solution

Answer

a. \(x = \frac{7}{9}\); b. \(x = -\frac{70}{9}\).

Step 1: Simplify the Expression for Part (a)

Start by using the logarithmic identity for subtraction: \( \log a - \log b = \log \left( \frac{a}{b} \right) \). Apply this to the given equation \( \log x - \log (x+7) = -1 \). This simplifies to: \[\log \left( \frac{x}{x+7} \right) = -1. \]

Step 2: Eliminate the Logarithm for Part (a)

To eliminate the logarithm, rewrite the equation in exponential form. Since \( \log_{10} (\frac{x}{x+7}) = -1 \), this means: \[\frac{x}{x+7} = 10^{-1} = \frac{1}{10}.\]

Step 3: Solve for \(x\) in Part (a)

Cross-multiply the equation \( \frac{x}{x+7} = \frac{1}{10} \) to get \( 10x = x + 7 \). Simplify and solve for \(x\):\[10x - x = 7\] \[9x = 7\] \[x = \frac{7}{9}.\]

Step 1: Simplify the Expression for Part (b)

Again use the identity: \( \log a - \log b = \log \left( \frac{a}{b} \right) \). Substitute into the equation: \( \log x - \log (x+7) = 1 \) which becomes: \[\log \left( \frac{x}{x+7} \right) = 1.\]

Step 2: Eliminate the Logarithm for Part (b)

Convert the logarithmic equation to an exponential one. From \( \log_{10} (\frac{x}{x+7}) = 1 \), we get: \[\frac{x}{x+7} = 10^1 = 10.\]

Step 3: Solve for \(x\) in Part (b)

Cross-multiply \( \frac{x}{x+7} = 10 \) and solve: \( x = 10(x + 7) \). Expanding and simplifying gives: \[x = 10x + 70\] \[-9x = 70\] \[x = -\frac{70}{9}.\]

Key Concepts

Logarithmic Identity

A logarithmic identity is a fundamental rule used to simplify expressions involving logarithms. One such important identity is the subtraction identity:

• \( \log a - \log b = \log \left( \frac{a}{b} \right) \)

This identity helps us to combine two logarithmic terms into a single logarithm by dividing the arguments of the logs. It's particularly useful in solving logarithmic equations, as it allows us to condense terms and make the equation more manageable.
In our exercise, both parts (a) and (b) utilize the log subtraction identity to simplify the expressions. For example, in part (a), \( \log x - \log (x+7) \) becomes \( \log \left( \frac{x}{x+7} \right) \). Understanding and applying this identity correctly is crucial to progressing to the next steps of solving logarithmic equations.

Exponential Form

Transforming logarithmic expressions to exponential form is an essential step in solving logarithmic equations. The basic principle to remember is:

• If \( \log_b a = c \), then in exponential form, it reads as \( a = b^c \).

This conversion is straightforward but immensely powerful. It removes the logarithmic function, effectively transforming the problem into a simpler algebraic equation.
In part (a) of our example, the equation \( \log \left( \frac{x}{x+7} \right) = -1 \) is transformed to \( \frac{x}{x+7} = 10^{-1} \) using the principle that \( 10^{-1} = \frac{1}{10} \). Similarly, in part (b), \( \log \left( \frac{x}{x+7} \right) = 1 \) converts to \( \frac{x}{x+7} = 10^1 = 10 \). This transformation is vital for clearing the log and tackling the algebra.

Cross-Multiplication

Cross-multiplication is a technique used to solve rational or fraction-based equations by eliminating the fractions. Once we have an equation in the form of \( \frac{a}{b} = \frac{c}{d} \), we can apply cross-multiplication as:

• \( a \cdot d = b \cdot c \)

This simplifies the equation to a more standardized algebraic form without denominators, making it easier to solve for the unknown variable.
In part (a) of the exercise, the equation \( \frac{x}{x+7} = \frac{1}{10} \) becomes \( 10x = x + 7 \) through cross-multiplication. Similarly, in part (b), \( \frac{x}{x+7} = 10 \) transforms to \( x = 10(x + 7) \). This step is critical for isolating the variable and moving toward a solution.

Solving for x

The final goal in a logarithmic or algebraic equation is to isolate the variable, usually denoted as \( x \), to determine its value. Once the equation is simplified by cross-multiplying, we're left with a linear equation which is easier to solve.

• In part (a), \( 10x = x + 7 \) simplifies to \( 9x = 7 \), leading to \( x = \frac{7}{9} \).
• In part (b), \( x = 10x + 70 \) rearranges to \( -9x = 70 \), giving \( x = -\frac{70}{9} \).

Solving for \( x \) involves basic algebraic steps such as addition, subtraction, multiplication, and division. The solutions, \( \frac{7}{9} \) and \( -\frac{70}{9} \), are derived by isolating \( x \) on one side of the equation. Understanding each step ensures clarity and correctness in arriving at the answer.