Combining a rational function with a logarithm expression

Key takeaway: This expression combines a rational function with a scaled logarithm, so domain restrictions come from both parts.

Structure of the expression

The expression

$\frac{38}{25-x^2}-0.1\ln(x)$

combines a rational term with a logarithmic term. To understand it well, you must check the restrictions from both pieces.

Domain analysis

The rational part requires

$25-x^2\neq 0 \quad \Rightarrow \quad x\neq \pm 5.$

The logarithmic part requires

$x>0.$

When both conditions are combined, the domain is

$x>0 \text{ and } x\neq 5.$

The value $x=-5$ is already excluded by $x>0$, so the effective domain is all positive real numbers except $5$.

Why the expression is not easy to simplify

There is no algebraic cancellation between the rational term and the logarithmic term because they are fundamentally different types of functions. One is a rational function, and the other is a transcendental logarithmic function.

So the expression should be treated as a sum/difference of separate parts, not as something that can be combined into a single fraction or simplified by factor cancellation.

Behavior and interpretation

This kind of expression often appears in calculus, graphing, or modeling problems. The rational term has vertical asymptotes at $x=\pm 5$, while the logarithmic term is only defined for positive inputs and grows slowly as $x$ increases.

The negative sign in front of $0.1\ln(x)$ means the logarithmic contribution is subtracted. That affects the graph shape, but not the domain.

Key points to remember

• Check each term separately for restrictions.
• Combine restrictions using "and."
• Do not try to cancel unlike function types.
• Expect asymptotic behavior near $x=5$.

If you are asked to evaluate or graph this expression, always start with the domain before doing any numerical work.

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Pitfalls the pros know 👇 A common mistake is to apply only one restriction and forget the other. For example, some students remember that $\ln(x)$ needs $x>0$ but overlook the denominator zero at $x=5$. Others try to merge the terms into a single fraction and cancel something that does not actually factor. Rational functions and logarithms do not simplify together by ordinary algebraic cancellation, so each term must be checked on its own first.

What if the problem changes? If the expression changes to $\frac{38}{25-x^2}+0.1\ln(x)$, the domain is exactly the same: $x>0$ and $x\neq 5$. The sign in front of the logarithm changes the graph above or below the x-axis, but it does not change where the function is defined. If the rational denominator were changed to $16-x^2$, the excluded values would become $x=\pm 4$, and then the combined domain would be $x>0$ with $x\neq 4$.

Tags: combined domain, rational expression, logarithmic function