Question

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

Short answer

Explain. Answer: Yes, as $x\to \mathrm{\infty }$, the logarithm functions $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$ grow at comparable rates, because the limit of the ratio of the two functions is a constant value, which is independent of the bases $a$ and $b$.

Step-by-step solution

Express one logarithm in terms of another

To find a relationship between the two logarithmic functions, we can use the logarithmic conversion formula ${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$. So, we have: $$f(x)=\frac{g(x)}{{\log }_{b}a}$$

Analyze the limit of the ratio

Now, let's find the limit of the ratio between the two functions as $x\to \mathrm{\infty }$: $$\underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{g(x)}=\underset{x\to \mathrm{\infty }}{lim}\frac{\frac{g(x)}{{\log }_{b}a}}{g(x)}$$

Simplify the limit

Notice that the $g(x)$ terms in the numerator and denominator cancel each other out. So, we are left with the following limit: $$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{{\log }_{b}a}$$

Evaluate the limit

Since ${\log }_{b}a$ is a constant value and the limit of a constant value is the constant value itself, we obtain: $$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{{\log }_{b}a}=\frac{1}{{\log }_{b}a}$$

Conclusion

The limit of the ratio of the two logarithm functions is a constant value as $x\to \mathrm{\infty }$. This means that $f(x)$ and $g(x)$ grow at comparable rates as $x\to \mathrm{\infty }$, regardless of the bases $a$ and $b$.

Key concepts

Comparable Growth Rates

When studying mathematical functions, understanding how they grow as their variable approaches infinity is crucial. This concept, often referred to as growth rates, helps determine the behavior of functions as they expand or contract. Two functions are said to grow at comparable rates if their ratio approaches a constant value as the variable approaches infinity.
In the case of the logarithmic functions $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$, even though the bases $a$ and $b$ may differ, the core idea is that these logarithmic functions grow similarly as $x$ becomes very large. This is because logarithms are fundamentally about exponents that compress large values into smaller, more manageable numbers.
To determine comparable growth rates, you can take the limit of the ratio of two functions. If this limit results in a constant, especially as $x$ approaches infinity, then the two functions grow at comparable rates. This is evident in the above example, where the ratio of $f(x)$ to $g(x)$ resolves to a constant, showing their growth similarity despite different bases.

Logarithmic Conversion Formula

The logarithmic conversion formula is a powerful tool for transforming logarithms of different bases into one another. The conversion is made possible by the equation:
$${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$$
This formula operates on the principle that you can change the base of a logarithm by dividing by the logarithm of the new base with respect to the old base. This process maintains the logarithmic relationship between the numbers.
For example, by applying this conversion to the functions $f(x)={\log }_{a}x$ and $g(x)={\log }_{b}x$, it's possible to express $f(x)$ in terms of $g(x)$. This makes comparison straightforward, as you can analyze their growth by centering the discussion around a common base.
The formula's utility lies in its ability to standardize logarithmic expressions, making it easier to compute and compare functions across different logarithmic bases.

Limits of Functions

Limits are a fundamental concept in calculus that describe how a function behaves as its input approaches a specific value, often infinity. Understanding limits allows us to explore the behavior and trend of functions beyond finite boundaries.
In the context of comparing logarithmic functions, finding the limit of their ratio as $x\to \mathrm{\infty }$ helps in assessing the relative growth of each function. By knowing $$\underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{g(x)}$$, and specifically by simplifying this limit as we did earlier to $$\frac{1}{{\log }_{b}a}$$, we essentially say that the two functions behave similarly, growing at a similar pace as $x$ increases.
This concept of limits is not only confined to logarithmic functions but applies across various mathematical frameworks, offering insights into the behavior of sequences, series, and other diverse types of functions. In essence, it acts as a bridge between finite computations and infinite potential outcomes.