Question

Solve the inequality.\(-3 \log _5 x+6 \leq 9\)

Short answer

The solution to the inequality is \(x \leq 0.2\) with x > 0.

Step-by-step solution

Isolate the logarithmic expression

Start by isolating the logarithmic expression on one side of the inequality by subtracting 6 from both sides: \(-3 \log _5 x \leq 3\)

Remove the coefficient of the logarithmic term

Remove the coefficient of the logarithmic term on the left by dividing both sides of the inequality by -3. Remember to flip the inequality sign as you’re dividing by a negative number. This gives: \(\log _5 x \geq -1\)

Convert the logarithmic inequality into an exponential inequality

Utilize the basic definition of logarithms to convert this into an exponential inequality. In general, if \(\log _b a = c\), then this implies that \(b^c = a\). Using this rule gives: \(5^{-1} \geq x\) which simplifies to \(0.2 \geq x\), or alternatively, \(x \leq 0.2\).

Identify the solution set

This implies that any \(x \leq 0.2\) is a solution to the original inequality, where x is positive since log is defined only for positive numbers.

Key concepts

Solving Logarithmic Inequalities

When dealing with \textbf{logarithmic inequalities}, the process involves manipulating the inequality so that the logarithm is isolated on one side. This enables us to make use of the properties of logarithms to find the solution set.

To start, just like the given exercise \( -3 \log _5 x+6 \leq 9 \), one must initially simplify the inequality to isolate the logarithmic term. After subtracting 6 from both sides and dividing by -3, it's crucial to recall the rule that flipping the inequality sign is necessary when dividing by a negative. This transforms the inequality into \( \log _5 x \geq -1 \).

To determine the solution, we convert the logarithmic inequality into an exponential form, which is often more straightforward to comprehend. Using the defining property of logarithms, we find that \( x \leq 0.2 \), but we must also consider the domain of the logarithmic function, which is x must be positive. Therefore, the inequality's solution set includes all positive values of x up to and including 0.2.

Properties of Logarithms

The \textbf{properties of logarithms} are essential tools in solving logarithmic equations and inequalities. Among these properties, the most relevant for our exercise include the following:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n \log_b(m) \)

Additionally, understanding the relationship between logarithms and exponents is crucial: \( \log_b(a) = c \) implies \( b^c = a \).

These rules help us to break down complex logarithmic expressions into simpler parts, making it easier to solve the inequalities. For example, if we had a coefficient in front of the logarithm, such as in \( -3 \log_5(x) \), applying the power rule in reverse would allow us to rewrite it as \( \log_5(x^3) \) before solving.

Inequality Sign Rules

Working with inequalities requires an understanding of the \textbf{inequality sign rules}. An important aspect to note is the behavior of inequalities under certain operations:

• When both sides of an inequality are multiplied or divided by a positive number, the direction of the inequality remains unchanged.
• However, if you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign to maintain the inequality's truth.

Our exercise perfectly illustrates the second rule, where dividing by -3 requires flipping the inequality from \( \leq \) to \( \geq \). This is a critical step and a common source of mistakes for students. It is always useful to perform a quick check with a simple example to confirm the direction of the inequality.

Exponential Functions

Understanding \textbf{exponential functions} is pivotal when converting logarithmic inequalities into a more manageable form. An exponential function has the form \( y = b^x \), where b is a positive real number, and x is any real number.

In the context of logarithmic inequalities, once isolated, the logarithm can be converted to its exponential equivalent using the base of the logarithm and the property that \( \log_b(a) = c \) is equivalent to \( b^c = a \). For instance, take the equation \( \log _5 x = -1 \). By expressing this in terms of an exponential function, we get \( 5^{-1} = x \) or \( x = 0.2 \).

Recognizing this link is crucial when deciphering the logarithmic inequality and determining the set of x values that satisfy it. If the inequality were written with the variable x in the exponent, analyzing the function's growth would also help in finding the solution.