Question

Determine the equation of the inverse of \(y=\log _{2}\left(\log _{3} x\right)\).

Short answer

The inverse function is \(f^{-1}(x) = 3^{2^x}\).

Step-by-step solution

Understand the Given Function

The given function is \(y = \log_{2} (\log_{3} x)\). This is a composition of two logarithmic functions.

Swap the Variables

To find the inverse, start by swapping \(x\) and \(y\), yielding \(x = \log_{2} (\log_{3} y)\). This represents the inverse relationship.

Isolate the Inner Logarithm

Rearrange the equation to isolate the inner logarithm by taking the exponential base 2 of both sides: \[ 2^x = \log_{3} y \].

Solve for y

Exponentiate both sides again using base 3 to solve for \(y\): \[ y = 3^{2^x} \].

Write the Inverse Function

Now, write the inverse function: \(f^{-1}(x) = 3^{2^x}\).

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are written as \(\backslash log_{b}(x)\), where \(b\) is the base. Understanding logarithms is crucial for solving many algebraic problems.
Logarithms help us determine the power to which a base must be raised to produce a given number. For example, if \(2^3 = 8\), then \(\backslash log_{2}(8) = 3\). This relationship makes logarithms very useful in various fields like science, engineering, and finance.
When working with logarithmic functions, remember these important properties:

• \(\backslash log_{b}(mn) = \backslash log_{b}(m) + \backslash log_{b}(n)\)
• \(\backslash log_{b}(m/n) = \backslash log_{b}(m) - \backslash log_{b}(n)\)
• \(\backslash log_{b}(m^n) = n \backslash log_{b}(m)\)

Logarithmic functions have a domain of all positive real numbers and a range of all real numbers. Understand these properties to easily tackle problems involving logarithms.

Composition of Functions

Composition of functions involves applying one function to the results of another. For instance, in the function \(y = \backslash log_{2} (\backslash log_{3} x)\), the inner function is \(\backslash log_{3} x\) and the outer function is \(\backslash log_{2} y\).
The composition is written as \(f(g(x))\), meaning you first apply \(g\) and then \(f\). Here's how to handle composed functions:

• Identify the inner and outer functions.
• Substitute the inner function into the outer one.
• Apply the chain rule if needed (in calculus).

Composing functions are essential in various branches of mathematics, ensuring you understand how different functions interact and transform values.

Exponential Functions

Exponential functions are the reverse of logarithmic functions. They are written as \(a^x\), where \(a\) is a positive constant called the base. Exponential functions model growth and decay processes, such as population growth or radioactive decay.
Key points about exponential functions:

• They grow faster than polynomial functions.
• The base \(a\) determines the rate of growth or decay.
• Their range is all positive real numbers, but the domain is all real numbers.

Inverting exponential functions gives us logarithmic functions. To solve \(y = a^x\) for \(x\), you use the logarithm \(x = \backslash log_{a}(y)\).
Being familiar with both exponential and logarithmic functions makes it easier to work through problems like finding the inverse of composed functions, such as in the given exercise.