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An exponential function is one in which the variable appears in the exponent. They have the general form:

• \( f(x) = a^{x} \)

where \( a \) is a positive constant, not equal to 1.
These functions show growth or decay behavior very quickly and are commonly used to model real-world situations like population growth or radioactive decay.
For example, our function here, \( f(x) = 10^x \), is an exponential function where the base of the exponent is 10.
The base, 10, indicates that the function grows by a factor of 10 for every unit increase in \( x \).
This leads to rapid growth as \( x \) increases.
Understanding exponential functions is crucial for finding the difference quotient in this exercise because they directly define how \( f(x) \) changes as \( x \) changes.

Simplifying expressions involves rewriting them in their simplest form. This concept is crucial to solving problems efficiently and accurately.
Simplifying can include combining like terms, factoring, or using algebraic identities.
When dealing with the difference quotient for an exponential function like \( f(x) = 10^x \), simplification focuses on breaking down the terms properly:

• Write \( f(x + h) = 10^{x + h} \) as \( 10^x \cdot 10^h \)
• Insert this into the difference quotient formula: \( \frac{10^x \cdot 10^h - 10^x}{h} \)
• Factor out \( 10^x \) from the numerator to simplify to \( \frac{10^x(10^h - 1)}{h} \)

This process of simplification makes it much easier to interpret and work with the expression, especially when analyzing the behavior of the function.

The properties of exponents are vital for handling expressions that involve exponential functions. Here, we use them to simplify and manipulate exponentials:

• Product of Powers: When multiplying like bases, add the exponents:\[ a^m \cdot a^n = a^{m+n} \]
• Quotient of Powers: When dividing like bases, subtract the exponents:\[ \frac{a^m}{a^n} = a^{m-n} \]
• Power of a Power: Apply the exponent outside the parentheses to each term inside:\[ (a^m)^n = a^{m \cdot n} \]

In the context of our exercise, we directly apply the product of powers property. We rewrote \( 10^{x+h} = 10^x \cdot 10^h \) during simplification.
Also, recognizing that \( 10^x \) can be factored out effectively used these properties to simplify our expression further:\( \frac{10^x(10^h - 1)}{h} \).
These properties make working with exponential functions easier, ensuring calculations remain accurate and manageable.