Question

If \(f(x)=\log _{a} x,\) show that \(-f(x)=\log _{1 / a} x\)

Short answer

-\( f(x) = \log_{1/a} x \)

Step-by-step solution

Understand the given function

Given the function definition: \[ f(x) = \log_{a} x \]

Apply the property of logarithms involving base inversion

Recall the logarithm property that states: \[ \log_{a} x = \frac{\log_{b} x}{\log_{b} a} \] where \(b\) is any positive number. In this case, set \(b = \frac{1}{a}\)

Substitute in the property

Substitute \( b = \frac{1}{a} \) into the property: \[ \log_{a} x = \frac{\log_{1/a} x}{\log_{1/a} a} \]

Simplify using the definition of logarithms

Use the fact that \( \log_{1/a} a = -1 \): \[ \frac{\log_{1/a} x}{-1} = -\log_{1/a} x \]

Equate the expressions

Equate the simplified expression with the original definition: \[ \log_{a} x = -\log_{1/a} x \]Therefore, \[ -f(x) = -\log_{a} x = \log_{1/a} x \]

Key concepts

logarithm base change

Changing the base of a logarithm can make some calculations easier. The logarithm base change formula is: \(\text{log}_{a} b = \frac{\text{log}_{c} b}{\text{log}_{c} a} \). This formula is powerful because it allows us to convert logarithms from one base to another. For example, if you want to change the base of \(\text{log}_{a} x\) to base \(\frac{1}{a}\), you substitute in the base change formula:\ \(\text{log}_{a} x = \frac{\text{log}_{1/a} x}{\text{log}_{1/a} a}\). This allows you to work with a new base that's more convenient for your problem. Note that \(\text{log}_{1/a} a = -1\), which simplifies the calculation.

logarithm inversion

Logarithm inversion refers to the concept that when you invert the base of a logarithm, the sign changes. If \(\text{log}_{a} x\) is a logarithm with base \a\, then inverting the base results in \(\text{log}_{1/a} x\). According to the properties of logarithms, \(\text{log}_{a} x = -\text{log}_{1/a} x\). This is seen in our given function \(-f(x) = \text{log}_{1/a} x\). This inversion is useful when dealing with logarithmic equations because it allows you to convert between inverted bases effortlessly.

logarithmic identities

Logarithmic identities simplify and connect various logarithmic expressions. One essential identity is the change-of-base formula mentioned earlier. Other useful identities include: \(\text{log}_{a} (xy) = \text{log}_{a} x + \text{log}_{a} y\) and \(\text{log}_{a} \frac{x}{y} = \text{log}_{a} x - \text{log}_{a} y \). These identities help in expanding and simplifying complex logarithmic expressions. For the given problem, using the identity \(\text{log}_{1/a} a = -1\) is crucial for the solution, as it allows for the transformation from \(\text{log}_{a} x \) to \(-\text{log}_{1/a} x \), completing the inversion process.