Question

Solve each equation. $$ 5^{x}=5^{-6} $$

Short answer

The solution to the equation is \(x = -6\).

Step-by-step solution

Understand the equation

The given equation is \(5^{x} = 5^{-6}\). Both sides of the equation have the same base of 5. This similarity allows us to equate the exponents directly.

Set the exponents equal

Since \(5^{x} = 5^{-6}\), the equation can be simplified by setting the exponents equal to each other: \(x = -6\).

Key concepts

Solving Exponential Equations

Exponential equations involve unknowns in the exponent, such as in the equation given: \(5^x = 5^{-6}\). The goal is to solve for the unknown exponent.

To solve such equations:

• Make sure both sides of the equation have the same base.
• When the bases are the same, you can set the exponents equal to each other.

In our example, both sides have the base 5. Thus, the equation can be simplified to \(x = -6\) by directly equating the exponents. This makes solving exponential equations straightforward if you can get the same base on both sides.

Equating Exponents

Equating exponents is a fundamental step when solving exponential equations.

Here's a closer look:

• When the bases are identical, you can ignore the bases and focus only on the exponents.
• For instance, in \(5^x = 5^{-6}\), since the bases (both 5) are the same, equate the exponents: \(x = -6\).

It's important to check that the bases are exactly the same first. If the bases are different, you may need to convert them to the same base or use other techniques.
In general, for equations like \(a^m = a^n\), the solution is to set \(m = n\).

Base Properties

Understanding the properties of bases in exponential equations is key to solving them efficiently.

Here are some properties to keep in mind:

• An exponential function with the same base allows you to set exponents equal if the resulting equality holds.
• An important fact is that if \(a^m = a^n\), then \(m = n\), provided \(a\) is a positive real number other than 1.

Returning to our example: \(5^x = 5^{-6}\)
The base 5 remains unchanged, allowing us to solve for the exponent directly. This simplifies our work to finding values that make the exponents equal.

It’s also important to know that any non-zero base raised to an exponent will never equal zero. For example, \(a^x = 0\) has no solution for non-zero \(a\) and real \(x\).