How to Find the Inverse of a Logarithmic Function

TL;DR

To find the inverse of the function f(x) = 4log₃(x + 7) - 9, start by replacing f(x) with y and switching x and y. Isolate the logarithmic term, exponentiate both sides, and then solve for y, resulting in the inverse function f⁻¹(x) = 3^((x + 9)/4) - 7.

Transcript

we're being asked to find the inverse so let's go ahead and work it out the first step in this problem is to replace f of x with y so y is equal to 4 times the log base three of x plus seven minus nine the second step is to switch x and y so instead of y we'll have x equals four times the log base three of y plus seven minus nine the third step req... Read More

Key Insights

• ❣️ Begin by replacing f(x) with y and switching x and y in the function equation.
• 🍉 Isolate the logarithmic term to simplify the inverse calculation.
• 😀 Exponentiate both sides to eliminate the logarithmic function and find y in terms of x.

Questions & Answers

Q: How do you begin finding the inverse of a logarithmic function?

To begin finding the inverse, replace f(x) with y and switch x and y in the function equation.

Q: What is the importance of isolating the logarithmic term in the inverse calculation?

Isolating the logarithmic term helps in solving for y by applying exponential properties to simplify the equation.

Q: Why is it necessary to exponentiate both sides in the inverse calculation process?

Exponentiating both sides helps eliminate the logarithmic function and find the expression for y in terms of x.

Q: How do you finalize the inverse calculation and present the answer correctly?

The final step involves writing the answer with correct notation, representing the inverse of the logarithmic function.

Summary & Key Takeaways

• Replace f(x) with y and switch x and y to get x = 4*log3(y) + 7 - 9.

• Isolate the log term and solve for y to get y = 3^(x+9)/4 - 7.

• Finally, write the answer with correct notation as f inverse of x = 3^(x+9)/4 - 7.