If [tex]$5 \log _3 3 x=15$[/tex], what is the value of x?

Divide both sides of the equation by 5: lo g 3 ​ 3 x = 3 .
Rewrite in exponential form: 3 x = 3 3 = 27 .
Divide by 3 to solve for x : x = 3 27 ​ .
Simplify to find the value of x : 9 ​ .

Explanation

Understanding the problem We are given the equation 5 lo g 3 ​ 3 x = 15 and we want to find the value of x .

Isolating the logarithm First, we divide both sides of the equation by 5 to isolate the logarithm: lo g 3 ​ 3 x = 5 15 ​ .

Simplifying the equation Simplifying the right side, we get: lo g 3 ​ 3 x = 3 .

Converting to exponential form Now, we rewrite the logarithmic equation in exponential form: 3 x = 3 3 .

Simplifying the exponent Simplifying the right side, we have: 3 x = 27 .

Solving for x Finally, we divide both sides by 3 to solve for x : x = 3 27 ​ .

Finding the value of x Simplifying the fraction, we find the value of x : x = 9 .


Examples
Logarithms are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and calculating the loudness of sound in decibels. Understanding how to solve logarithmic equations is essential for these applications. For example, if we know the intensity of an earthquake is 1000 times greater than the reference intensity, we can use logarithms to find its magnitude on the Richter scale: R = lo g 10 ​ ( 1000 ) = 3 .

Answered by GinnyAnswer | 2025-07-03

To solve the equation 5 lo g 3 ​ 3 x = 15 , we isolate the logarithm, convert it to exponential form, and solve for x to find that x = 9 .
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Answered by Anonymous | 2025-07-04