What is the solution to $\log _5(x+30)=3$?
A. $x=-95$
B. $x=-5$
C. $x=5$
D. $x=95
Answer (2)
Rewrite the logarithmic equation in exponential form: x + 30 = 5 3 .
Calculate 5 3 : 5 3 = 125 .
Solve for x : x = 125 − 30 .
The solution is: 95 .
Explanation
Understanding the Problem We are given the equation lo g 5 ( x + 30 ) = 3 . Our goal is to find the value of x that satisfies this equation.
Converting to Exponential Form To solve for x , we need to rewrite the logarithmic equation in exponential form. The equation lo g 5 ( x + 30 ) = 3 is equivalent to x + 30 = 5 3 .
Calculating the Power Now, we calculate 5 3 . 5 3 = 5 × 5 × 5 = 125 . So, we have x + 30 = 125 .
Isolating x To isolate x , we subtract 30 from both sides of the equation: x = 125 − 30 .
Finding the Solution Finally, we calculate the value of x : x = 125 − 30 = 95 . Therefore, the solution to the equation is x = 95 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, measuring the intensity of sound in decibels, and determining the pH of a solution in chemistry. 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. Similarly, in finance, logarithmic functions are used to model growth and decay processes, such as compound interest and depreciation.
To solve lo g 5 ( x + 30 ) = 3 , convert it to exponential form to get x + 30 = 125 . After calculating and isolating x, we find that x = 95 . Thus, the correct answer is option D: x = 95 .
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