Question

If \(b\) is a positive real number other than 1 , then \(b^x=b^y\) if and only if

Short answer

In the equation \(b^x = b^y\), \(x\) is equal to \(y\), if \(b\) is a positive real number which is not 1.

Step-by-step solution

Understanding the given

Understanding the provided exercise. It's given that \(b\) is a positive real number other than 1, and \(b^x = b^y\). The task is to determine the relationship between \(x\) and \(y\). From the properties of exponents, it's known that if the bases are the same, then they must have the same exponents.

Applying the property of exponents

Applying the property of exponents. In this case, the rule implies that, in the equation \(b^x = b^y\), \(x\) must be equal to \(y\). Because if \(x\) was not equal to \(y\), then \(b^x\) would not be equal to \(b^y\). This property is true only if \(b\) is positive real number and not equal to 1.

Key concepts

Exponent Equations

Exponent equations, like the one presented in the exercise, reveal the fundamental nature of exponents in algebra. They often come in the form of \( b^x = b^y \) where \( b \) is the base and \( x \) and \( y \) are the exponents. To solve these equations, one of the key principles to apply is that if the bases are the same and the equation is valid, the exponents must be equal too. This means that for \( b^x = b^y \) to hold true, it must be that \( x = y \).

This might seem straightforward, but it is pivotal for ensuring you don't accidentally apply this rule to situations where \( b \) is not a constant or when working with bases that are not equal. For example, in contrast, the equation \(b^x = c^y\) does not imply that \( x = y \) unless \( b = c \) as well. Understanding and correctly applying this rule is vital for solving more complex algebraic equations involving exponents.

Algebraic Properties

Algebraic properties are the backbone of solving equations and simplifying expressions in algebra. These properties include the commutative, associative, distributive properties, and those specific to exponents. In the context of the exercise, we focus on the properties of exponents. One such property is that when two exponential expressions with the same base are equal, as in \( b^x = b^y \), we can deduce that their exponents must be identical assuming the base \( b \) is a positive real number and not equal to one.

Other useful exponent properties are multiplication \( b^x \cdot b^y = b^{x + y} \), division \( \frac{b^x}{b^y} = b^{x - y}\), and power of a power \( (b^x)^y = b^{x \cdot y} \) rules. Being familiar with these helps in understanding the underlying logic that leads to the conclusion that \( x = y \) when \( b^x = b^y \) given that \( b \) meets the necessary condition.

Positive Real Numbers

In the given exercise, the condition that \( b \) is a positive real number other than 1 is crucial. Positive real numbers are all the numbers greater than zero that you can find on the number line. They are essential in mathematics not only for their arithmetic properties but also because they maintain order and have predictable behaviors with operations, especially with exponents.

For exponents, the base being a positive real number except 1 ensures that powers of the base can't be equal unless the exponents are the same. This wouldn't be the case if \( b \) were negative, zero, or one. For example, if \( b \) were negative, then \( b^x \) could equal \( b^y \) for certain integer values of \( x \) and \( y \) where one is even and the other odd, because negative numbers raised to even exponents yield positive results. If \( b \) were 1, then \( b^x \) would always equal \( b^y \) as any number to the power of any exponent equals one if the number is one. Therefore, \( b \) being a positive real number other than 1 is a key stipulation for the steps taken in the solution to be valid.