Question

If \(x\) satisfies \(\log _{5}(2 x+3)<\log _{5} 7, \operatorname{tnen} x\) lies in : (a) \(\left(-\frac{3}{2}, 2\right)\) (b) \(\left(\frac{3}{2}, 2\right)\) (c) \((0, \infty)\) (d) \((0,2)\)

Short answer

a) \(\left(-\frac{3}{2}, 2\right)\) b) \(\left(-\frac{3}{2}, 2\right]\) c) \(\left[-\frac{3}{2}, 2\right)\) d) \(\left[-\frac{3}{2}, 2\right]\) Answer: a) \(\left(-\frac{3}{2}, 2\right)\)

Step-by-step solution

Eliminate the logarithm on both sides of the inequality

Since both sides of the inequality are in log base 5 form, we can eliminate the logarithm by raising both sides as a power of the base 5. This results in the inequality: \(5^{\log_{5}(2x+3)} < 5^{\log_{5}7}\) which simplifies to: \(2x+3 < 7\)

Solve the inequality for x

Now we will solve the inequality to find the possible values of x: \(2x+3 < 7\) Subtract 3 from both sides of the inequality: \(2x < 4\) Lastly, divide both sides by 2: \(x < 2\) So, we see that x is less than 2.

Choose the correct interval for x

We now must check the given options to see which one satisfies the condition we found, namely that \(x < 2\). The interval that contains x should not have an upper limit greater than or equal to 2 since x must be less than 2. Comparing the given options with the condition, the only suitable option is: (a) \(\left(-\frac{3}{2}, 2\right)\) So, the correct answer is option (a) \(\left(-\frac{3}{2}, 2\right)\).

Key concepts

Inequality Solving

Solving inequalities involves finding the range of values that satisfy a given inequality. It is similar to solving equations, but with extra care given to the direction of the inequality sign:
* If you add or subtract the same number to both sides of an inequality, the inequality remains balanced.
* When multiplying or dividing both sides by a positive number, the inequality's direction does not change.
* However, if multiplying or dividing by a negative number, the inequality sign flips direction.

For example, in the exercise provided, we start by simplifying the inequality from \(2x+3 < 7\) and use basic arithmetic steps:
1. Subtract 3 to isolate terms with x, leading to \(2x < 4\).
2. Divide by 2 to solve for x, giving us \(x < 2\).

This is a simple linear inequality, revealing the value range where x satisfies the original inequality condition.

Logarithm Rules

Logarithms are incredibly useful in solving equations where the unknown variable is in an exponent. Understanding basic log rules simplifies complex expressions:
* The logarithm of a product, \(\log_b(M \cdot N) = \log_b M + \log_b N\), splits into a sum of logs.
* The logarithm of a quotient, \ \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\, becomes a difference of logs.
* The logarithm of a power, \ \log_b(M^n) = n \log_b M\, brings down the exponent in front of the logarithm.

In our particular problem, the property used is that if \(\log_b(M) < \log_b(N)\), then \(M < N\), given the logs have the same base and the numbers are positive. This allowed us to transform \(\log_5(2x+3) < \log_5(7)\) into \(2x+3 < 7\) easily, which simplifies the process of solving the inequality.

Interval Notation

Interval notation is a concise way of displaying the solution set for an inequality, indicating precisely which values an inequality holds true for:
* The notation \(a, b\) indicates all numbers between a and b, not including a and b themselves.
* If a or b are included in the interval, square brackets \[a, b\] are used instead of parentheses.
* Infinite intervals use \(a, \infty\) or \(-\infty, b\), showing values go on indefinitely in a positive or negative direction.

In the exercise, the correct answer is \((-\frac{3}{2}, 2)\) indicating x is greater than \(-\frac{3}{2}\) and less than 2. A well-chosen interval succinctly communicates the set of solutions, letting us know exactly the values that meet the inequality conditions.