Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, use the definition of a logarithm to solve the equation. \(-8 \log _{9} x=16\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{81}\).

Step 1: Isolate the logarithm

First, isolate the logarithmic expression by dividing both sides of the equation by -8:\[-8 \log_{9} x = 16 \]Dividing both sides by -8 yields:\[\log_{9} x = \frac{16}{-8} = -2 \]

Step 2: Convert from logarithmic form to exponential form

Using the definition of a logarithm, \(\log_b a = c\) is equivalent to \(b^c = a\). In this case, convert \(\log_9 x = -2\) to exponential form:\[9^{-2} = x\]

Step 3: Evaluate the exponential expression

Calculate \(9^{-2}\):\[9^{-2} = \frac{1}{9^2} = \frac{1}{81}\]So, \(x = \frac{1}{81}\).

Key Concepts

Definition of a Logarithm

Logarithms are incredibly useful in mathematics for solving equations involving exponential growth or decay. The definition of a logarithm states that if you have an equation in the form of \( \log_b a = c \), it means that the base \( b \) raised to the power \( c \) will give you \( a \). In simpler terms, the logarithm asks, "To what power must \( b \) be raised, to result in \( a \)?"
This definition helps transform complicated equations into manageable problems. If you can understand 'logarithms', many mathematical puzzles become much easier to solve.
By applying the definition of a logarithm, we can easily switch between logarithmic and exponential expressions, an essential skill when simplifying or solving equations.

Isolate the Logarithm

When you're faced with an equation involving a logarithm, often the first task is to isolate the logarithmic term. This means getting the logarithm on one side of the equation by itself. In our original exercise, we had \( -8 \log_{9} x = 16 \).
To isolate \( \log_{9} x \), divide both sides of the equation by -8:

• Divide: \( -8 \log_{9} x \/ -8 = 16 \/ -8 \)
• Simplify: \( \log_{9} x = -2 \)

Isolating the logarithm makes it easier to transform the equation into an exponential form. This step is crucial as it sets the stage for applying the definition we've just learned.

Convert Logarithmic to Exponential Form

After isolating the logarithm, the next step is to convert it into an exponential form using the definition of a logarithm.
In the exercise, the logarithmic equation was simplified to \( \log_{9} x = -2 \).
According to the definition, this can be converted into an exponential equation:

• The base is \( 9 \).
• The exponent is \( -2 \).
• Therefore, \( 9^{-2} = x \).

Converting to exponential form allows us to directly solve for the unknown variable \( x \). Practice this conversion to enhance your problem-solving skills whenever you encounter logarithmic equations.

Evaluate Exponential Expressions

Once the equation is in exponential form, the next task is to evaluate the expression to find the solution. In our example, we need to evaluate \( 9^{-2} \).
This can be calculated as follows:

• Recognize \( 9^{-2} \) as a fraction: \( \frac{1}{9^2} \).
• Calculate \( 9^2 \), which is \( 81 \).
• Thus, \( 9^{-2} = \frac{1}{81} \).

Therefore, \( x = \frac{1}{81} \). Evaluating exponential expressions accurately is the final step in solving logarithmic equations and confirms your solution. Taking careful steps ensures that your calculations are correct, thereby solving the problem effectively.