Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, use the definition of a logarithm to solve the equation. \(10-4 \ln (9-8 x)=6\)

Step-by-Step Solution

Answer

\(x = \frac{9 - e}{8}\).

Step 1: Isolate the logarithm term

Begin by isolating the term containing the logarithm. Start with the equation \(10 - 4 \ln(9 - 8x) = 6\). Subtract 10 from both sides to get:\[-4 \ln(9 - 8x) = -4\]

Step 2: Simplify the equation

Divide both sides of the equation by \(-4\) to obtain:\[\ln(9 - 8x) = 1\]

Step 3: Exponentiate to eliminate the logarithm

Recall that if \(\ln(a) = b\), then \(e^b = a\). In this equation, raise \(e\) to the power of both sides to remove the natural logarithm:\[9 - 8x = e^1\] which simplifies to \[9 - 8x = e\].

Step 4: Solve for x

Now, solve for \(x\) by subtracting 9 from both sides, giving:\[-8x = e - 9\]. Next, divide by \(-8\):\[x = \frac{9 - e}{8}\].

Key Concepts

Solving Logarithms

Logarithms allow us to solve equations involving exponential terms by providing a means to "undo" the exponential process. In an equation like the one in the given exercise, \(10 - 4 \ln(9 - 8x) = 6\), the first step to solve for the variable \(x\) is to manipulate the equation to isolate the logarithmic term.

• Start by isolating \(\ln(9 - 8x)\) on one side of the equation. This involves clearing any additional numbers or coefficients attached to it.
• After isolating the logarithm, further simplify the equation if necessary, such as by dividing or multiplying the terms.

This process reduces the complexity of the equation and prepares it for the next stage, which involves removing the logarithm through exponentiation.

Exponential Functions

An exponential function is one where the variable appears in the exponent. These functions commonly arise when solving logarithmic equations after isolating and simplifying the logarithm. In our example, converting the equation \(\ln(9 - 8x) = 1\) to its exponential form involves the use of the inverse property of logarithms:

• If \(\ln(a) = b\), then it is equivalent to say \(e^b = a\).
• This reveals that the natural logarithm \(\ln\) and the exponential function \(e\) are inverse operations, meaning they can 'undo' each other.

Thus, raising \(e\) to both sides of the equation results in \(9 - 8x = e^1\), which simplifies to \(9 - 8x = e\).
Understanding this allows you to transition an equation from its logarithmic form to an algebraic form that is generally easier to solve.

Natural Logarithm (ln)

The natural logarithm, denoted as \(\ln\), is a logarithm to the base of the natural constant \(e\approx 2.718\). It is pivotal in mathematics for simplifying equations that contain exponential growth models. In solving logarithmic equations, particularly those involving \(\ln\), it is critical to convert the logarithmic expression into an exponential one for complete evaluation.
In the original exercise, converting \(\ln(9-8x)=1\) into its exponential form as \(e^1 = 9-8x\) was a key step. Here’s why \(\ln(a)=b\) translates to \(e^b = a\):

• \(\ln\) and the exponential function \(e^x\) are inverses; this allows interchangeability, facilitating simplification in solving equations.
• The natural logarithm models many real-world situations, such as population dynamics and radioactive decay, thus making it essential for interpreting and solving such phenomena.

By mastering \(\ln\), students can navigate between logarithmic expressions and their exponential equivalents, crucial for proficiency in higher-level mathematics.