Question

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

Short answer

Since \(f\left(\frac{1}{x}\right) = -\log_{a}(x) = -f(x)\), we have \(f\left(\frac{1}{x}\right) = -f(x)\).

Step-by-step solution

- Understand the given function

Given the function is:ewline\[ f(x) = \log_{a}(x) \] This function represents the logarithm of \(x\) with base \(a\).

- Substitute \( \frac{1}{x} \) into the function

We need to find the value of \[ f \left( \frac{1}{x} \right) \] Substitute \( \frac{1}{x} \) into the function \[ f\left(\frac{1}{x}\right) = \log_{a} \left( \frac{1}{x} \right) \]

- Apply the properties of logarithms

Use the property of logarithms that \( \log_{a} \left( \frac{1}{x} \right) = -\log_{a} (x) \), which states the logarithm of the reciprocal of a number is the negative logarithm of the number:\[ f \left( \frac{1}{x} \right) = -\log_{a} (x) \]

- Conclude the result

Since \[ -\log_{a} (x) = -f (x) \], we can write:\[ f\left(\frac{1}{x}\right) = -f(x) \]

Key concepts

logarithms

Logarithms are mathematical functions that help us to simplify and solve exponential equations. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, if we have \(\text{log}_{b}(a) = c\), it means that \(b^{c} = a\). Here, \(b\) is the base, \(a\) is the number, and \(c\) is the logarithm. Understanding logarithms is crucial in many areas of math and science, especially when dealing with exponential growth or decay.

logarithm of a reciprocal

The logarithm of a reciprocal involves taking the logarithm of the fraction \(\frac{1}{x}\). According to the properties of logarithms, this can be simplified using a specific rule: the logarithm of a reciprocal is equal to the negative logarithm of the original number. Mathematically, this is represented as \(\text{log}_{a}(\frac{1}{x}) = -\text{log}_{a}(x)\). This property can be extremely helpful when solving logarithmic equations involving reciprocals.

algebraic proofs

Algebraic proofs involve logical steps and properties of algebra to demonstrate a mathematical statement is true. To prove the properties of functions or equations, you often manipulate the expressions using known algebraic rules and properties. For instance, in the given problem, we used the properties of logarithms to prove the relationship \(f\big(\frac{1}{x}\big) = -f(x)\). By substituting \(\frac{1}{x} \) into the logarithmic function and using the property of reciprocal logarithms, we reached the desired conclusion.

properties of logarithms

Properties of logarithms are essential tools that make calculations more straightforward. Some key properties include:

• \(\text{log}_{a}(x \times y) = \text{log}_{a}(x) + \text{log}_{a}(y)\)
• \(\text{log}_{a}(\frac{x}{y}) = \text{log}_{a}(x) - \text{log}_{a}(y)\)
• \(\text{log}_{a}(x^{y}) = y \times \text{log}_{a}(x)\)
• \(\text{log}_{a}(\frac{1}{x}) = -\text{log}_{a}(x)\)

These properties allow simplification and easy manipulation of logarithmic expressions, making it easier to solve complex equations.