Find the inverse function of f.
f(x) = log5(x − 2)
Can someone explain to me how do i get this answer like this???? Thank you!!!
the answer is:
=ln(x-2)/ln(5)
f-1(x)= 5^x+2
To find the inverse function of f(x) = log5(x − 2), we need to switch the roles of x and y and solve for y.
Step 1: Start with the equation f(x) = log5(x − 2).
Step 2: Switch the roles of x and y so that the equation becomes x = log5(y − 2).
Step 3: Rewrite the equation in exponential form. In exponential form, log5(y − 2) = x is equivalent to 5^x = y − 2.
Step 4: Solve for y by adding 2 to both sides of the equation. This gives us 5^x = y − 2 + 2, which simplifies to 5^x = y.
Step 5: Replace y with f-1(x) to represent the inverse function. Therefore, f-1(x) = 5^x.
So, the inverse function of f(x) = log5(x − 2) is f-1(x) = 5^x.
Note: In the answer you provided, ln(x-2)/ln(5), it seems like there was an error in the calculation or notation as it does not correspond to the correct inverse function. The correct inverse function is f-1(x) = 5^x.
To find the inverse function of f(x) = log5(x - 2), we follow these steps:
Step 1: Start with the original function f(x) = log5(x - 2).
Step 2: Replace f(x) with y. The equation becomes y = log5(x - 2).
Step 3: Swap the x and y values. The equation becomes x = log5(y - 2).
Step 4: Rewrite the equation exponentiating both sides with the base 5 since we are dealing with a logarithm to the base 5. This will eliminate the logarithm on the right side. We get 5^x = y - 2.
Step 5: Solve for y. We add 2 to both sides of the equation to isolate y. The equation becomes 5^x + 2 = y.
So, the inverse function of f(x) = log5(x - 2) is f^-1(x) = 5^x + 2.
Note: The term ln(x) is the natural logarithm, which uses the base e (approximately 2.71828). In this case, since the original function is using log base 5, we keep the answer 5^x + 2 instead of using the natural logarithm.