What is the approximate value of $x$ in the equation below?
[tex]\log _5 15=x+3[/tex]
Answer (2)
Isolate x in the equation: x = lo g 5 15 − 3 .
Calculate lo g 5 15 ≈ 1.6826 .
Substitute the value back into the equation: x ≈ 1.6826 − 3 .
The approximate value of x is − 1.317 .
Explanation
Understanding the Problem We are given the equation lo g 5 15 = x + 3 and asked to find the approximate value of x .
Isolating x To solve for x , we need to isolate it on one side of the equation. We can do this by subtracting 3 from both sides: x = lo g 5 15 − 3
Calculating the Logarithm Now we need to find the value of lo g 5 15 . Using a calculator, we find that lo g 5 15 ≈ 1.6826 .
Finding the Approximate Value of x Substitute this value back into the equation for x : x ≈ 1.6826 − 3 x ≈ − 1.3174
Final Answer The approximate value of x is − 1.3174 . Comparing this to the given options, the closest value is − 1.317 .
Examples
Logarithms are used in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, measuring the loudness of sound in decibels, and determining the pH of a chemical solution. Understanding how to solve logarithmic equations is essential in these fields. For example, if we know the magnitude of an earthquake is 6 on the Richter scale, we can use logarithms to determine the amplitude of the seismic waves.
To find x in the equation lo g 5 15 = x + 3 , we isolate x and calculate lo g 5 15 using the change of base formula. After calculations, we find that x ≈ − 1.315 . Therefore, the approximate value of x is − 1.315 .
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