Consider the equation $\log _5(x+5)=x^2$. What are the approximate solutions of the equation? Check all that apply.

A. $x \approx-0.93$
B. $x=0$
C. $x \approx 0.87$
D. $x \approx 1.06$

Evaluate lo g 5 ​ ( x + 5 ) and x 2 for each given option.
For x ≈ − 0.93 , lo g 5 ​ ( − 0.93 + 5 ) ≈ 0.8721 and ( − 0.93 ) 2 ≈ 0.8649 , so it's a solution.
For x = 0 , lo g 5 ​ ( 0 + 5 ) = 1 and ( 0 ) 2 = 0 , so it's not a solution.
For x ≈ 0.87 , lo g 5 ​ ( 0.87 + 5 ) ≈ 1.0997 and ( 0.87 ) 2 ≈ 0.7569 , so it's not a solution.
For x ≈ 1.06 , lo g 5 ​ ( 1.06 + 5 ) ≈ 1.1195 and ( 1.06 ) 2 ≈ 1.1236 , so it's a solution.
The approximate solutions are x ≈ − 0.93 , x ≈ 1.06 ​ .

Explanation

Problem Analysis We are given the equation lo g 5 ​ ( x + 5 ) = x 2 and asked to find the approximate solutions from the given options: x ≈ − 0.93 , x = 0 , x ≈ 0.87 , x ≈ 1.06 . We will evaluate each option to see if it satisfies the equation.

Checking x = -0.93 Let's check the first option, x ≈ − 0.93 . We need to calculate lo g 5 ​ ( − 0.93 + 5 ) and ( − 0.93 ) 2 and see if they are approximately equal. From the tool, we have lo g 5 ​ ( − 0.93 + 5 ) ≈ 0.8721 and ( − 0.93 ) 2 ≈ 0.8649 . These values are very close, so x ≈ − 0.93 is a solution.

Checking x = 0 Next, we check the option x = 0 . We need to calculate lo g 5 ​ ( 0 + 5 ) and ( 0 ) 2 . We have lo g 5 ​ ( 5 ) = 1 and 0 2 = 0 . Since 1  = 0 , x = 0 is not a solution.

Checking x = 0.87 Now, we check the option x ≈ 0.87 . We need to calculate lo g 5 ​ ( 0.87 + 5 ) and ( 0.87 ) 2 . From the tool, we have lo g 5 ​ ( 0.87 + 5 ) ≈ 1.0997 and ( 0.87 ) 2 ≈ 0.7569 . These values are not close, so x ≈ 0.87 is not a solution.

Checking x = 1.06 Finally, we check the option x ≈ 1.06 . We need to calculate lo g 5 ​ ( 1.06 + 5 ) and ( 1.06 ) 2 . From the tool, we have lo g 5 ​ ( 1.06 + 5 ) ≈ 1.1195 and ( 1.06 ) 2 ≈ 1.1236 . These values are very close, so x ≈ 1.06 is a solution.

Final Answer Therefore, the approximate solutions of the equation lo g 5 ​ ( x + 5 ) = x 2 are x ≈ − 0.93 and x ≈ 1.06 .


Examples
Logarithmic equations are used in various fields such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and modeling population growth or decay. For instance, in finance, logarithmic scales are used to represent stock market indices or investment returns, allowing for easier comparison of percentage changes. Understanding how to solve these equations helps in making informed decisions based on the data presented in logarithmic form.

Answered by GinnyAnswer | 2025-07-05