Question

Show that $f(x)={\log }_{a}x$ and $g(x)=$ ${\log }_{b}x,$ where $a>1$ and $b>1,$ grow at a comparable rate as $x\to \mathrm{\infty }$

Short answer

Answer: Yes, the functions $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$ grow at a comparable rate as $x\to \mathrm{\infty }$.

Step-by-step solution

Write the given functions and their ratio.

We have two functions, $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$. Their ratio is defined by $\frac{f(x)}{g(x)}=\frac{{\log }_{a}x}{{\log }_{b}x}$.

Use the change of base formula for logarithms.

We can rewrite the ratio using the change of base formula: ${\log }_{x}A=\frac{{\log }_{y}A}{{\log }_{y}x}$. In our case, we have $\frac{{\log }_{a}x}{{\log }_{b}x}=\frac{{\log }_{c}x}{{\log }_{c}a}÷\frac{{\log }_{c}x}{{\log }_{c}b}$, where $c$ is any common base for both logarithms.

Simplify the expression.

Using the properties of fractions, we can write the expression as $\frac{{\log }_{c}x}{{\log }_{c}a}\cdot \frac{{\log }_{c}b}{{\log }_{c}x}$. We observe that ${\log }_{c}x$ cancels out, leaving the simplified expression: $\frac{{\log }_{c}b}{{\log }_{c}a}$.

Take the limit of the simplified expression as $x$ approaches infinity.

Now, we analyze the limit as $x\to \mathrm{\infty }$: $\underset{x\to \mathrm{\infty }}{lim}\frac{{\log }_{c}b}{{\log }_{c}a}$. Since ${\log }_{c}b$ and ${\log }_{c}a$ are constants, the limit is also constant and equal to the given simplified expression: $\underset{x\to \mathrm{\infty }}{lim}\frac{{\log }_{c}b}{{\log }_{c}a}=\frac{{\log }_{c}b}{{\log }_{c}a}$. Since the limit of the ratio of the functions as $x$ approaches infinity is a constant and not equal to 0 or infinity, we can conclude that $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$ grow at a comparable rate as $x\to \mathrm{\infty }$.

Key concepts

Comparable Growth Rate in Logarithmic Functions

In mathematics, understanding how functions grow, especially as they approach infinity, can provide valuable insight into their behavior. The concept of comparable growth rate essentially states whether two functions increase at similar rates as their variable grows large.

For logarithmic functions like $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$, comparing their growth rates involves examining the ratio $\frac{f(x)}{g(x)}=\frac{{\log }_{a}x}{{\log }_{b}x}$. If this ratio approaches a constant as $x$ goes to infinity (excluding 0 or infinity), the functions are said to grow comparably.

• The ratio itself simplifies by employing the change of base formula.
• The outcome, a constant, confirms comparable growth.

A constant result suggests both ${\log }_{a}x$ and ${\log }_{b}x$ escalate at consistent rates, adding deeper understanding to their behavior across vast values of $x$.

The Limit of a Function: Understanding the Concept

Limits play a crucial role in determining the behavior of functions as they move towards a particular point or as they extend towards infinity. The limit gives us a sense of the value that a function approaches as the input gets infinitely large or infinitesimally small.

In our example, to determine if $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$ grow comparably, we compute $\underset{x\to \mathrm{\infty }}{lim}\frac{{\log }_{a}x}{{\log }_{b}x}$.

• If this limit results in a finite non-zero constant, it states that while each function increases, they do so by similar proportions.
• When limits yield constants, they provide a powerful conclusion about the relationship between functions.

In this case, since the limit as $x$ approaches infinity results in a constant that is neither zero nor infinite, we ascertain that the functions indeed exhibit comparable growth at infinity.

Using the Change of Base Formula in Logarithms

The change of base formula is a fundamental tool for manipulating logarithms, enabling us to convert a logarithm from one base to another.

This formula is crucial when calculating the growth rate of functions like ${\log }_{a}x$ and ${\log }_{b}x$. Given any base $c$, the formula is expressed as ${\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}$.

• This transformation helps simplify comparisons and calculations.
• It allows different bases to be unified, facilitating a general comparison between functions like ${\log }_{a}x$ and ${\log }_{b}x$.

Through this equation, the exercise simplifies the ratio of two logarithms, $\frac{{\log }_{a}x}{{\log }_{b}x}$, into a constant, underscoring comparable growth. The change of base formula is therefore indispensable in deducing function behavior, particularly in understanding complex expressions and helping clarify the comparable growth in logarithmic functions.