The $x$-intercept of the graph of $f(x)=3 \log (x-5)+2$ is:
A. $10^{-2 / 3}-5$
B. $10^{-2 / 3}+5$
C. $10^{2 / 3}+5$
D. $10^{2 / 3}-5$
Answer (2)
Set f ( x ) = 0 to find the x-intercept.
Solve the equation 3 lo g ( x − 5 ) + 2 = 0 for x .
Isolate the logarithmic term: lo g ( x − 5 ) = − 3 2 .
Solve for x : x = 1 0 − 3 2 + 5 .
The x-intercept is 1 0 − 3 2 + 5 .
Explanation
Understanding the problem We are given the function f ( x ) = 3 lo g ( x − 5 ) + 2 and we want to find the x -intercept. The x -intercept is the value of x when f ( x ) = 0 .
Setting up the equation To find the x -intercept, we set f ( x ) = 0 and solve for x : 3 lo g ( x − 5 ) + 2 = 0
Isolating the logarithm Subtract 2 from both sides: 3 lo g ( x − 5 ) = − 2
Further isolating the logarithm Divide both sides by 3: lo g ( x − 5 ) = − 3 2
Exponentiating both sides Exponentiate both sides with base 10: 1 0 l o g ( x − 5 ) = 1 0 − 3 2 x − 5 = 1 0 − 3 2
Solving for x Add 5 to both sides to solve for x : x = 1 0 − 3 2 + 5
Final Answer Therefore, the x -intercept is 1 0 − 3 2 + 5 .
Examples
Finding the x-intercept of a logarithmic function is useful in various real-world scenarios. For example, in analyzing the decay of radioactive substances, the x-intercept can represent the time at which the substance reaches a certain negligible level. Similarly, in financial modeling, it can indicate when an investment reaches a break-even point. Understanding logarithmic functions and their intercepts helps in making informed decisions in science, engineering, and finance.
To find the x -intercept of f ( x ) = 3 lo g ( x − 5 ) + 2 , we set the function equal to zero and solve for x , leading us to the result x = 1 0 − 3 2 + 5 . Therefore, the correct answer is option B. This shows how to manipulate logarithmic equations to find intercepts on a graph.
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