Problem 75 - College Algebra

Question

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. $$ \log (-0.7 x-9)=1+5 \log (5) $$

Step-by-Step Solution

Answer

The solution is approximately \( x = -44358.4287 \).

Step 1: Rewrite the Equation

First, rewrite the equation in a more standard format. We have:\[ \log(-0.7x - 9) = 1 + 5\log(5) \] We know that the right side can be broken down further, so let's focus on that next.

Step 2: Simplify the Right Side

We'll rewrite the right-side expression, \[ 1 + 5\log(5) \].Calculate \( 5\log(5) \) first.Given \( \log(5) \approx 0.69897 \), Calculate \( 5 \times 0.69897 = 3.49485 \), Then add 1 to it to get,\[ 1 + 3.49485 = 4.49485 \].So, the equation is now: \[ \log(-0.7x - 9) = 4.49485 \].

Step 3: Equate the Log Equations

Since \( \log(a) = b \) implies \( a = 10^b \), We have:\[ -0.7x - 9 = 10^{4.49485} \]Calculate \( 10^{4.49485} \) using a calculator.

Step 4: Calculate the Exponential Value

Compute the value \( 10^{4.49485} \) using a calculator, which gives approximately,\[ 10^{4.49485} \approx 31041.9001 \].Now we have the equation:\[ -0.7x - 9 = 31041.9001 \].

Step 5: Solve for x

Now solve for \( x \): First, isolate \( -0.7x \) by adding 9 to each side:\[ -0.7x = 31041.9001 + 9 \],\[ -0.7x = 31050.9001 \].Then, divide by \(-0.7\) to find \( x \):\[ x = \frac{31050.9001}{-0.7} \].Using a calculator, compute this:\[ x \approx -44358.4287 \].

Step 6: Final Answer

Round \( x \approx -44358.4287 \) to the nearest ten-thousandth: \( x = -44358.4287 \).

Key Concepts

Using a calculator efficiently

When solving logarithmic equations, a calculator can be invaluable, especially when dealing with complex expressions. Calculators help simplify calculations involving logarithms and exponentials, ensuring accuracy and efficiency. In equations like \( \log(-0.7x - 9) = 1 + 5\log(5) \), a scientific or graphing calculator aids in:

• Determining logarithmic values, such as \( \log(5) \approx 0.69897 \).
• Simplifying expressions like \( 5\log(5) \) or computing powers, such as \( 10^{4.49485} \).

By using these functions, students can quickly navigate through steps, thus focusing more on understanding the process rather than getting bogged down with arithmetic. Regular practice with a calculator helps in building confidence, especially while preparing for exams or solving homework problems.

Understanding exponential functions in logarithms

Exponential functions appear frequently in solving logarithmic equations. When we have equations like \( \log(a) = b \), it directly translates to the exponential form \( a = 10^b \). This shows the deep relationship between logarithms and exponential functions.In the given problem, once the logarithmic equation is simplified, it translates to an exponential operation:

• From \( \log(-0.7x - 9) = 4.49485 \) to \(-0.7x - 9 = 10^{4.49485} \).

Understanding this transition is key, as it allows you to shift the problem from a logarithmic to a more manageable exponential perspective. Students often find it easier to solve once it is rephrased into an exponential format, which deals directly with multiplication and powers rather than logarithmic properties.

Step-by-step in solving equations

Solving equations, especially those involving logarithms, requires a systematic approach. Here’s how you can break down the process:1. **Rewrite the Equation:** Reorganize complex logs into a manageable form, like converting \( 1 + 5\log(5) \) into a simplified number, as shown in the example.2. **Simplify Both Sides:** Compute components on both sides. For instance, simplify the right side to get \( \log(-0.7x - 9) = 4.49485 \).3. **Use Exponential Properties:** Translate the logarithmic form into an exponential equation. This helps in eliminating the log, making it easier to isolate \( x \).4. **Solve for the Variable:** After forming the exponential equation \(-0.7x - 9 = 31041.9001 \), solve for \( x \) by isolating it and using basic algebra, like adding or dividing.By following these structured steps, you gradually narrow down to the solution. This method not only helps in pinpointing errors but also reinforces understanding and confidence in tackling similar problems.